Mathematical Relations Az

Mathematical relations azFrom Wikipedia, the free encyclopedia

Contents

1 Accessibility relation 11.1 Description of Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Basic Review of (Propositional) Modal Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 The Four Types of the 'Accessibility Relation' in Formal Semantics . . . . . . . . . . . . . . . . . 41.4 Philosophical Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.5 Computer Science Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Allegory (category theory) 72.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Regular categories and allegories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 Allegories of relations in regular categories . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2.2 Maps in allegories, and tabulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.3 Unital allegories and regular categories of maps . . . . . . . . . . . . . . . . . . . . . . . 82.2.4 More sophisticated kinds of allegory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Alternatization 93.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.1.1 Alternating bilinear map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.1.2 Alternating bilinear form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.1.3 Alternating multilinear form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.1.4 Alternatization of a bilinear map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4 Ancestral relation 124.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.2 Relationship to transitive closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

i

ii CONTENTS

4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5 Antisymmetric relation 145.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

6 Asymmetric relation 166.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

7 Better-quasi-ordering 187.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187.3 Simpson’s alternative definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187.4 Major theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

8 Bidirectional transformation 208.1 Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208.2 Vocabulary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208.3 Examples of implementations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

9 Bijection 229.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

9.2.1 Batting line-up of a baseball team . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239.2.2 Seats and students of a classroom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

9.3 More mathematical examples and some non-examples . . . . . . . . . . . . . . . . . . . . . . . . 249.4 Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249.5 Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249.6 Bijections and cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249.7 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259.8 Bijections and category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

CONTENTS iii

9.9 Generalization to partial functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269.10 Contrast with . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269.11 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269.12 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269.13 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279.14 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

10 Bijection, injection and surjection 2810.1 Injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2810.2 Surjection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2910.3 Bijection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3010.4 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3010.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

10.5.1 Injective and surjective (bijective) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3110.5.2 Injective and non-surjective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3110.5.3 Non-injective and surjective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3110.5.4 Non-injective and non-surjective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

10.6 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3210.7 Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3210.8 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3210.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3210.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

11 Homogeneous relation 3311.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

11.1.1 Is a relation more than its graph? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3411.1.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

11.2 Special types of binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3411.2.1 Difunctional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

11.3 Relations over a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3611.4 Operations on binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

11.4.1 Complement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3811.4.2 Restriction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3811.4.3 Algebras, categories, and rewriting systems . . . . . . . . . . . . . . . . . . . . . . . . . 39

11.5 Sets versus classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3911.6 The number of binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3911.7 Examples of common binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4011.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4011.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4011.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4111.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

iv CONTENTS

12 Cointerpretability 4312.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4312.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

13 Commutative property 4413.1 Common uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4413.2 Mathematical definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4513.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

13.3.1 Commutative operations in everyday life . . . . . . . . . . . . . . . . . . . . . . . . . . . 4513.3.2 Commutative operations in mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . 4513.3.3 Noncommutative operations in everyday life . . . . . . . . . . . . . . . . . . . . . . . . . 4613.3.4 Noncommutative operations in mathematics . . . . . . . . . . . . . . . . . . . . . . . . . 47

13.4 History and etymology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4713.5 Propositional logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

13.5.1 Rule of replacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4813.5.2 Truth functional connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

13.6 Set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4813.7 Mathematical structures and commutativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4813.8 Related properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

13.8.1 Associativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4913.8.2 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

13.9 Non-commuting operators in quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 5013.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5013.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5113.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

13.12.1 Books . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5113.12.2 Articles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5213.12.3 Online resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

14 Comparability 5314.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5314.2 Comparability graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5314.3 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5314.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5314.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

15 Composition of relations 5515.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5515.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5515.3 Join: another form of composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5615.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5615.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

CONTENTS v

15.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

16 Congruence relation 5716.1 Basic example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5716.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5716.3 Relation with homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5816.4 Congruences of groups, and normal subgroups and ideals . . . . . . . . . . . . . . . . . . . . . . . 58

16.4.1 Ideals of rings and the general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5916.5 Universal algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5916.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5916.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

17 Contour set 6017.1 Formal definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

17.1.1 Contour sets of a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6117.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

17.2.1 Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6117.2.2 Economic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

17.3 Complementarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6217.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6317.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6317.6 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

18 Coreflexive relation 6418.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

19 Demonic composition 6519.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6519.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

20 Dense order 6620.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6620.2 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6620.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6620.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

21 Dependence relation 6721.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6721.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

22 Dependency relation 6822.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

23 Homogeneous relation 70

vi CONTENTS

23.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7023.1.1 Is a relation more than its graph? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7123.1.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71





24 Directed set 8024.1 Equivalent definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8024.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8024.3 Contrast with semilattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8124.4 Directed subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8224.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8224.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8224.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

25 Equality (mathematics) 8325.1 Etymology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8325.2 Types of equalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

25.2.1 Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8325.2.2 Equalities as predicates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8325.2.3 Congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8325.2.4 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8325.2.5 Equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

25.3 Logical formalizations of equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8425.4 Logical formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8425.5 Some basic logical properties of equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8425.6 Relation with equivalence and isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8525.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8625.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

CONTENTS vii

26 Equipollence (geometry) 8726.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8726.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

27 Equivalence class 8927.1 Notation and formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8927.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9027.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9027.4 Graphical representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9127.5 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9127.6 Quotient space in topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9127.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9127.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9227.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9227.10Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

28 Equivalence relation 9428.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9428.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9428.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

28.3.1 Simple example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9428.3.2 Equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9528.3.3 Relations that are not equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

28.4 Connections to other relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9528.5 Well-definedness under an equivalence relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9628.6 Equivalence class, quotient set, partition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

28.6.1 Equivalence class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9628.6.2 Quotient set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9628.6.3 Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9628.6.4 Equivalence kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9628.6.5 Partition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

28.7 Fundamental theorem of equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9728.8 Comparing equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9728.9 Generating equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9728.10Algebraic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

28.10.1 Group theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9828.10.2 Categories and groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9928.10.3 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

28.11Equivalence relations and mathematical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9928.12Euclidean relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10028.13See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10028.14Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

viii CONTENTS

28.15References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10128.16External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

29 Euclidean relation 10329.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10329.2 Relation to transitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10329.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

30 Exceptional isomorphism 10430.1 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

30.1.1 Finite simple groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10430.1.2 Groups of Lie type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10430.1.3 Alternating groups and symmetric groups . . . . . . . . . . . . . . . . . . . . . . . . . . 10430.1.4 Cyclic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10530.1.5 Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10630.1.6 Coxeter groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

30.2 Lie theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10730.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10730.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10730.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

31 Fiber (mathematics) 10931.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

31.1.1 Fiber in naive set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10931.1.2 Fiber in algebraic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

31.2 Terminological variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10931.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

32 Finitary relation 11132.1 Informal introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11132.2 Relations with a small number of “places” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11232.3 Formal definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11232.4 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11332.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11332.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11332.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11432.8 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

33 Foundational relation 11533.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11533.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

34 Homogeneous relation 11634.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

CONTENTS ix

34.1.1 Is a relation more than its graph? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11734.1.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117





35 Hypostatic abstraction 12635.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12735.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12735.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

36 Idempotence 12836.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

36.1.1 Unary operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12836.1.2 Idempotent elements and binary operations . . . . . . . . . . . . . . . . . . . . . . . . . . 12936.1.3 Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

36.2 Common examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12936.2.1 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12936.2.2 Formal languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13036.2.3 Idempotent ring elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13036.2.4 Other examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

36.3 Computer science meaning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13036.3.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

36.4 Applied examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13136.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13136.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13136.7 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13236.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

37 Idempotent relation 13337.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

x CONTENTS

38 Intransitivity 13438.1 Intransitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13438.2 Antitransitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13438.3 Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13538.4 Occurrences in preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13538.5 Likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13538.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13638.7 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

39 Inverse relation 13739.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

39.1.1 Inverse relation of a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13739.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13739.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13839.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

40 Inverse trigonometric functions 13940.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

40.1.1 Etymology of the arc- prefix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13940.2 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

40.2.1 Principal values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13940.2.2 Relationships between trigonometric functions and inverse trigonometric functions . . . . . 14040.2.3 Relationships among the inverse trigonometric functions . . . . . . . . . . . . . . . . . . . 14040.2.4 Arctangent addition formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

40.3 In calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14140.3.1 Derivatives of inverse trigonometric functions . . . . . . . . . . . . . . . . . . . . . . . . 14140.3.2 Expression as definite integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14240.3.3 Infinite series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14240.3.4 Indefinite integrals of inverse trigonometric functions . . . . . . . . . . . . . . . . . . . . 144

40.4 Extension to complex plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14540.4.1 Logarithmic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

40.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14740.5.1 General solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14740.5.2 In computer science and engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

40.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14940.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14940.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

41 Near sets 15341.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15541.2 Nearness of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15741.3 Generalization of set intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

CONTENTS xi

41.4 Efremovič proximity space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15741.5 Visualization of EF-axiom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15841.6 Descriptive proximity space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15841.7 Proximal relator spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16041.8 Descriptive δ -neighbourhoods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16141.9 Tolerance near sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16241.10Tolerance classes and preclasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

41.10.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16341.11Nearness measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16441.12Near set evaluation and recognition (NEAR) system . . . . . . . . . . . . . . . . . . . . . . . . . 16541.13Proximity System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16541.14See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16641.15Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16641.16References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16741.17Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

42 Partial equivalence relation 17242.1 Properties and applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17242.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

42.2.1 Euclidean parallelism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17242.2.2 Kernels of partial functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17342.2.3 Functions respecting equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . 173

42.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17342.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

43 Partial function 17443.1 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17443.2 Total function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17543.3 Discussion and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

43.3.1 Natural logarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17543.3.2 Subtraction of natural numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17543.3.3 Bottom element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17543.3.4 In category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17543.3.5 In abstract algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

43.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17643.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

44 Partially ordered set 17744.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17844.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17844.3 Extrema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17844.4 Orders on the Cartesian product of partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . 179

xii CONTENTS

44.5 Sums of partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17944.6 Strict and non-strict partial orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18044.7 Inverse and order dual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18044.8 Mappings between partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18044.9 Number of partial orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18144.10Linear extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18144.11In category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18244.12Partial orders in topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18244.13Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18244.14See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18244.15Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18344.16References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18344.17External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

45 Preorder 18445.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18445.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18545.3 Uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18545.4 Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18645.5 Number of preorders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18645.6 Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18745.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18745.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

46 Prewellordering 18846.1 Prewellordering property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

46.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18846.1.2 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188


47 Sequential composition 19047.1 Essential features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19047.2 Mathematics of processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

47.2.1 Parallel composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19147.2.2 Communication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19147.2.3 Sequential composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19147.2.4 Reduction semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19147.2.5 Hiding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19247.2.6 Recursion and replication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19247.2.7 Null process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

47.3 Discrete and continuous process algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

CONTENTS xiii

47.4 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19247.5 Current research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19247.6 Software implementations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19347.7 Relationship to other models of concurrency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19347.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19347.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19447.10Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

48 Propositional function 19548.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

49 Quasi-commutative property 19649.1 Applied to matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19649.2 Applied to functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19649.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19749.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

50 Quasitransitive relation 19850.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19850.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19850.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19850.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19850.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

51 Quotient by an equivalence relation 20051.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20051.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20051.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20051.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

52 Rational consequence relation 20252.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20252.2 Uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

52.2.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20252.2.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

52.3 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20352.4 Rational consequence relations via atom preferences . . . . . . . . . . . . . . . . . . . . . . . . . 203

52.4.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20452.5 The representation theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

52.5.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20452.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

53 Reduct 20553.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

xiv CONTENTS

53.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20553.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

54 Reflexive closure 20654.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20654.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20654.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

55 Reflexive relation 20755.1 Related terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20755.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20755.3 Number of reflexive relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20855.4 Philosophical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20955.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20955.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21055.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21055.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

56 Relation algebra 21156.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

56.1.1 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21156.2 Expressing properties of binary relations in RA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21256.3 Expressive power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

56.3.1 Q-Relation Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21356.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21356.5 Historical remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21456.6 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21456.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21456.8 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21456.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21456.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

57 Relation construction 21657.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

58 Representation (mathematics) 21758.1 Representation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21758.2 Other examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

58.2.1 Graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21758.2.2 Order theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21758.2.3 Polysemy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218


CONTENTS xv

59 Semiorder 22059.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22059.2 Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22059.3 Other results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22259.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22259.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22359.6 Additional reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

60 Separoid 22460.1 The axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22460.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22460.3 The basic lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22560.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

61 Sequential composition 22661.1 Essential features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22661.2 Mathematics of processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

61.2.1 Parallel composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22761.2.2 Communication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22761.2.3 Sequential composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22761.2.4 Reduction semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22761.2.5 Hiding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22861.2.6 Recursion and replication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22861.2.7 Null process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

61.3 Discrete and continuous process algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22861.4 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22861.5 Current research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22861.6 Software implementations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22961.7 Relationship to other models of concurrency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22961.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22961.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23061.10Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

62 Series-parallel partial order 23162.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23262.2 Forbidden suborder characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23262.3 Order dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23262.4 Connections to graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23362.5 Computational complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23362.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23462.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23462.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

xvi CONTENTS

63 Surjective function 23563.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23663.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23663.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

63.3.1 Surjections as right invertible functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23763.3.2 Surjections as epimorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23863.3.3 Surjections as binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23963.3.4 Cardinality of the domain of a surjection . . . . . . . . . . . . . . . . . . . . . . . . . . . 23963.3.5 Composition and decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23963.3.6 Induced surjection and induced bijection . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

63.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23963.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24063.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

64 Symmetric closure 24164.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24164.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24164.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

65 Symmetric relation 24265.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

65.1.1 In mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24265.1.2 Outside mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

65.2 Relationship to asymmetric and antisymmetric relations . . . . . . . . . . . . . . . . . . . . . . . 24365.3 Additional aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24365.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

66 Ternary equivalence relation 24566.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24566.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

67 Ternary relation 24667.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

67.1.1 Binary functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24667.1.2 Cyclic orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24667.1.3 Betweenness relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24667.1.4 Congruence relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24667.1.5 Typing relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

67.2 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

68 Tolerance relation 24868.1 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

69 Total order 249

CONTENTS xvii

69.1 Strict total order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24969.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25069.3 Further concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

69.3.1 Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25069.3.2 Lattice theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25069.3.3 Finite total orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25169.3.4 Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25169.3.5 Order topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25169.3.6 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25169.3.7 Sums of orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

69.4 Orders on the Cartesian product of totally ordered sets . . . . . . . . . . . . . . . . . . . . . . . . 25269.5 Related structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25269.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25269.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25269.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

70 Total relation 25470.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25470.2 Properties and related notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25470.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25470.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

71 Transitive closure 25671.1 Transitive relations and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25671.2 Existence and description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25671.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25771.4 In graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25771.5 In logic and computational complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25771.6 In database query languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25871.7 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25871.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25871.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25971.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

72 Transitive relation 26072.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26072.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26072.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

72.3.1 Closure properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26172.3.2 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26172.3.3 Properties that require transitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

72.4 Counting transitive relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

xviii CONTENTS

72.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26172.6 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

72.6.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26272.6.2 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

72.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

73 Trichotomy (mathematics) 26373.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26373.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

74 Unimodality 26574.1 Unimodal probability distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

74.1.1 Other definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26774.1.2 Uses and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26774.1.3 Gauss’ inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26774.1.4 Vysochanskiï–Petunin inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26774.1.5 Mode, median and mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26774.1.6 Skewness and kurtosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268

74.2 Unimodal function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26874.3 Other extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26874.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26974.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

75 Weak ordering 27075.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27175.2 Axiomatizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

75.2.1 Strict weak orderings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27175.2.2 Total preorders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27275.2.3 Ordered partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27275.2.4 Representation by functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

75.3 Related types of ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27375.4 All weak orders on a finite set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

75.4.1 Combinatorial enumeration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27375.4.2 Adjacency structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

75.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27575.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

76 Well-founded relation 27676.1 Induction and recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27676.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27776.3 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27776.4 Reflexivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27876.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278

CONTENTS xix

77 Well-order 27977.1 Ordinal numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27977.2 Examples and counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280

77.2.1 Natural numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28077.2.2 Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28077.2.3 Reals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280

77.3 Equivalent formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28177.4 Order topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28177.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28277.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282

78 Well-quasi-ordering 28378.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28378.2 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28378.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28378.4 Wqo’s versus well partial orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28478.5 Infinite increasing subsequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28478.6 Properties of wqos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28478.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28578.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28578.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28578.10Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 286

78.10.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28678.10.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29378.10.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296

Chapter 1

Accessibility relation

In modal logic, an accessibility relation is a binary relation, written as R between possible worlds.

1.1 Description of Terms

A 'statement' in logic refers to a sentence (with a subject, predicate, and verb) that can be true or false. So, 'Theroom is cold' is a statement because it contains a subject, predicate and verb, and it can be true that 'the room is cold'or false that 'the room is cold.'Generally, commands, beliefs and sentences about probabilities aren't judged as true or false. 'Inhale and exhale' istherefore not a statement in logic because it is a command and cannot be true or false, although a person can obeyor refuse that command. 'I believe I can fly or I can't fly' isn't taken as a statement of truth or falsity, because beliefsdon't say anything about the truth or falsity of the parts of the entire 'and' or 'or' statement and therefore the entire'and' or 'or' statement.A 'possible world' is any possible situation. In every case, a 'possible world' is contrasted with an actual situation.Earth one minute from now is a 'possible world.' The earth as it actually is also a 'possible world.' Hence the oddityof and controversy in contrasting a 'possible' world with an 'actual world' (earth is necessarily possible). In logic,'worlds’ are described as a non-empty set, where the set could consist of anything, depending on what the statementsays.'Modal Logic' is a description of the reasoning in making statements about 'possibility' or 'necessity.' 'It is possiblethat it rains tomorrow' is a statement in modal logic, because it is a statement about possibility. 'It is necessary thatit rains tomorrow' also counts as a statement in modal logic, because it is a statement about 'necessity.' There are atleast six logical axioms or principles that show what people mean whenever they make statements about 'necessity' or'possibility' (described below). For a detailed explanation on modal logic, see here.As described in greater detail below:Necessarily p means that p is true at every 'possible world' w such that R(w∗, w).

Possibly p means that p is true at some possible world w such that R(w∗, w) .'Truth-Value' is whether a statement is true or false. Whether or not a statement is true, in turn, depends on themeanings of words, laws of logic, or experience (observation, hearing, etc.).'Formal Semantics’ refers to the meaning of statements written in symbols. The sentence (□p ∨ □q) → □(p ∨ q)

, for example, is a statement about 'necessity' in 'formal semantics.' It has a meaning that can be represented by thesymbol R .

The 'accessibility relation' is a relationship between two 'possible worlds.' More preciselyplease clarify definition, the'accessibility relation' is the idea that modal statements, like 'it’s possible that it rains tomorrow,' may not take thesame truth-value in all 'possible worlds.' On earth, the statement could be true or false. By contrast, in a planet wherewater is non-existent, this statement will always be false.Due to the difficulty in judging if a modal statement is true in every 'possible world,' logicians have derived certainaxioms or principles that show on what basis any statement is true in any 'possible world.' These axioms describing

1

https://en.wikipedia.org/wiki/Modal_logic

https://en.wikipedia.org/wiki/Binary_relation

https://en.wikipedia.org/wiki/Possible_world

http://videolectures.net/ssll09_gore_iml/

https://en.wikipedia.org/wiki/Truth-value

2 CHAPTER 1. ACCESSIBILITY RELATION

the relationship between 'possible worlds’ is the 'accessibility relation' in detail.Put another way, these modal axioms describe in detail the 'accessibility relation,' R between two 'worlds.' Thatrelation,R symbolizes that from any given 'possible world' some other 'possible worlds’ may be accessible, and othersmay not be.The 'accessibility relation' has important uses in both the formal/theoretical aspects of modal logic (theories about'modal logic'). It also has applications to a variety of disciplines including epistemology (theories about how peopleknow something is true or false), metaphysics (theories about reality), value theory (theories about morality andethics), and computer science (theories about programmatic manipulation of data).

1.2 Basic Review of (Propositional) Modal Logic

The reasoning behind the 'accessibility relation' uses the basics of 'propositional modal logic' (see modal logic for adetailed discussion). 'Propositional modal logic' is traditional propositional logic with the addition of two key unaryoperators:□ symbolizes the phrase 'It is necessary that...'♢ symbolizes the phrase 'It is possible that...'These operators can be attached to a single sentence to form a new compound sentence.For example, □ can be attached to a sentence such as 'I walk outside.' The new sentence would look like: □ 'I walkoutside.' The entire new sentence would therefore read: 'It is necessary that I walk outside.'But the symbol A can be used to stand for any sentence instead of writing out entire sentences. So any sentence suchas 'I walk outside' or 'I walk outside and I look around' are symbolized by A .Thus for any sentence A (simple or compound), the compound sentences □A and ♢A can be formed. Sentencessuch as 'It is necessary that I walk outside' or 'It is possible that I walk outside' would therefore look like: □ A ♢A .However, the symbols p , q can also be used to stand for any statement of our language. For example, p can standfor 'I walk outside' or 'I walk outside and I look around.' The sentence 'It is necessary that I walk outside' would looklike: □ q . The sentence 'It is possible that I walk outside' would look like: ♢ q .Six Basic Axioms of Modal Logic:There are at least six basic axioms or principles of almost all modal logics or steps in thinking/reasoning. The firsttwo hold in all regular modal logics, and the last holds in all normal modal logics.1st Modal Axiom:

• □p↔ ¬♢¬p (Duality)

The double arrow stands symbolizes 'if and only if,' 'necessary and sufficient' conditions. A 'necessary' condition issomething that must be the case for something else. Being literate, for instance, is a 'necessary' condition for readingabout the 'accessibility relation.' A 'sufficient condition' a condition that is good enough for something else. Beingliterate, for instance, is a 'sufficient' condition for learning about the accessibility relation.' In other words, it’s goodenough to be literate in order to learn about the 'accessibility relation,' however it may not be 'necessary' because therelation could be learned in different ways (like through speech). Aside from 'necessary and sufficient,' the doublearrow represents equivalence between the meaning of two statements, the statement to the left and the statement tothe right of the double arrow.The half square symbols before the diamond and p symbol in the sentence ' □p ↔ ¬♢¬p ' stand for 'it is not thecase, or 'not.'The p symbol stands for any statement such as 'I walk outside.' Therefore it could also stand for 'The apple is Red.'Example 1:The first principle says that any statement involving 'necessity' on the left side of the double arrow is equivalent to thestatement about the negation of 'possibility' on the right.So using the symbols and their meaning, the first modal axiom, □p ↔ ¬♢¬p could stand for: 'It’s necessary that Iwalk outside if and only if it’s not possible that it is not the case that I walk outside.'


https://en.wikipedia.org/wiki/Epistemology

https://en.wikipedia.org/wiki/Metaphysics

https://en.wikipedia.org/wiki/Value_theory

https://en.wikipedia.org/wiki/Computer_science


https://en.wikipedia.org/wiki/Propositional_logic

https://en.wikipedia.org/wiki/Regular_modal_logic

https://en.wikipedia.org/wiki/Normal_modal_logic

1.2. BASIC REVIEW OF (PROPOSITIONAL) MODAL LOGIC 3

And when I say that 'It’s necessary that I walk outside,' this is the same as saying that 'It’s not possible that it is notthe case that I walk outside.' Furthermore, when I say that 'It’s not possible that it is not the case that I walk outside,'this is the same as saying that 'It’s necessary that I walk outside.'Example 2:p stands for 'The apple is red.'So using the symbols and their meaning described above, the first modal axiom, □p ↔ ¬♢¬p could stand for: 'It’snecessary that the apple is red if and only if it’s not possible that it is not the case that the apple is red.'And when I say that 'It’s necessary that the apple is red,' this is the same as saying that 'It’s not possible that it is notthe case that the apple is red.' Furthermore, when I say that 'It’s not possible that it is not the case that the apple isred,' this is the same as saying that 'It’s necessary that the apple is red.'Second Modal Axiom:

• ♢p↔ ¬□¬p (Duality)

Example 1:The second principle says that any statement involving 'possibility' on the left side of the double arrow is the same asthe statement about the negation of 'necessity' on the right.p stands for 'Spring has not arrived.'Using the symbols and their meaning described above, the second modal axiom, ♢p ↔ ¬□¬p could stand for: 'It’spossible that Spring has not arrived if and only if it is not the case that it is necessary that it is not the case that Springhas not arrived.'Essentially, the second axiom says that any statement about possibility called 'X' is the same as a negation or denialof a different statement about necessity 'Y.' The statement about necessity shows the denial of the same originalstatement 'X.'The other axioms can be read and interpreted in the same way, by substituting letters p for any statement and followingthe reasoning. Brackets in a symbolized sentence mean that anything inside the brackets counts as a whole sentence.Any symbol before the brackets therefore applies to the sentence as a whole, not just the letters or an individualsentence.An arrow stands for “then” where the left statement before the arrow is the “if” of the entire sentence.Other Modal Axioms:* □(p ∧ q) ↔ (□p ∧□q)

* (□p ∨□q) → □(p ∨ q)

* □(p → q) → (□p → □q) (Kripke property)Most of the other axioms concerning the modal operators are controversial and not widely agreed upon. Here are themost commonly used and discussed of these:

(T) □p → p

(4) □p → □□p

(5) ♢p → □♢p

(B) p → □♢p

Here, "(T)","(4)","(5)", and "(B)" represent the traditional names of these axioms (or principles).According to the traditional 'possible worlds’ semantics of modal logic, the compound sentences that are formed outof the modal operators are to be interpreted in terms of quantification over possible worlds, subject to the relationof accessibility. A sentence like (□p ∨ □q) → □(p ∨ q) is to be interpreted as true or false in all or some 'possibleworlds.' In turn, the grounds on which the sentence is true (symmetry, transitive property, etc.) in all 'possible worlds’is the 'accessibility relation.'

https://en.wikipedia.org/wiki/Interpretation_(logic)


The relation of accessibility can now be defined as an (uninterpreted) relation R(w1, w2) that holds between 'possibleworlds’ w1 and w2 only when w2 is accessible from w1 .Additionally, to make things more specific, any formula, axiom like (□p ∨□q) → □(p ∨ q) can be translated into aformula of first-order logic through standard translation. That first-order logic formula or sentence makes the meaningof the boxes and diamonds in modal logic explicit.

1.3 The Four Types of the 'Accessibility Relation' in Formal Semantics

'Formal semantics’ studies the meaning of statements written in symbols. The sentence (□p ∨ □q) → □(p ∨ q) ,for example, is a statement about 'necessity' in 'formal semantics.' It has a meaning that can be represented by thesymbol R , where R takes the form of the 'necessity relation' described below.So, the 'accessibility relation,' R can take on at least four forms, that is, the 'accessibility relation' can be describedin at least four ways.Each type is either about 'possibility' or 'necessity' where 'possibility' and 'necessity' is defined as:

• (TS) Necessarily p means that p is true at every 'possible world' w such that R(w∗, w) .

• Possibly p means that p is true at some possible world w such that R(w∗, w) .

The four types of R will be a variation of these two general types. They will specify on what conditions a statementis true either in every possible world, or some possible. The four specific types of R are:Reflexive, or *Axiom (T) above:If R is reflexive, every world is accessible to itself. Reflexivity guarantees that any world at which A is true is aworld from which there is an accessible world at which A is true, and thus A is possible at worlds where it’s true,which isn't necessarily the case in worlds that aren't accessible to themselves. Without the reflexivity condition, Acan be necessary at a world where it’s false, if that world isn't accessible to itself; thus axiom T—that □A at a worldimplies A is true at that world—follows from reflexivity.Transitive, or *Axiom (4) above:If R is transitive, any world accessible to any world w′ accessible to world w is also accessible to w . Transitively,□A is true at a world w only when A is true at every world w′ accessible to w , including every world w′′

accessible to any w′ , and every world accessible to any w′′ , etc., so when □A is true at w , it’s also true at everyw′ and every w′′ , etc., which means □□A is also true at w , which is axiom 4.Euclidean or *Axiom (5) above:If R is euclidean, any two worlds accessible to a given world are accessible to each other. □♢A is true at a world w

if and only if, for every world w′ accessible to w , there is a world w′′ accessible to w′ at which A is true. If A

is true at a world w′ accessible to w , then if that world is accessible to every other world accessible to w , it willbe true that for every world accessible to w there is an accessible world ( w′ ) at which A is true, so ♢A is true atall worlds accessible to w . The euclidean property thus entails that ♢A implies □♢A , which is axiom 5.Symmetric or *Axiom (B) above:If R is symmetric, then if world w′ is accessible to world w , w is accessible to w′ . If A is true at w , then atevery w′ accessible to w , there is a world ( w ) accessible to w′ at which A is true, so A is possible at all w′ ,and thus it’s necessary at w that A is possible, which is axiom B.

1.4 Philosophical Applications

One of the applications of 'possible worlds’ semantics and the 'accessibility relation' is to physics. Instead of justtalking generically about 'necessity (or logical necessity),' the relation in physics deals with 'nomological necessity.'The fundamental translational schema (TS) described earlier can be exemplified as follows for physics:

• (TSN)P is nomologically necessary means thatP is true at all possible worlds that are nomologically accessible

https://en.wikipedia.org/wiki/First-order_logic

https://en.wikipedia.org/wiki/Standard_translation

https://en.wikipedia.org/wiki/Binary_relation#Relations_over_a_set

https://en.wikipedia.org/wiki/Euclidean_relation

https://en.wikipedia.org/wiki/Logical_truth

1.5. COMPUTER SCIENCE APPLICATIONS 5

from the actual world. In other words, P is true at all possible worlds that obey the physical laws of the actualworld.

The interesting thing to observe is that instead of having to ask, now, “Does nomological necessity satisfy the axiom(5)?", that is, “Is something that is nomologically possible nomologically necessarily possible?", we can ask instead:“Is the nomological accessibility relation euclidean?" And different theories of the nature of physical laws will resultin different answers to this question. (Notice however that if the objection raised earlier is true, each different theoryof the nature of physical laws would be 'possible' and 'necessary,' since the euclidean concept depends on the ideaabout 'possibility' and 'necessity'). The theory of Lewis, for example, is asymmetric. His counterpart theory alsorequires an intransitive relation of accessibility because it is based on the notion of similarity and similarity is generallyintransitive. For example, a pile of straw with one less handful of straw may be similar to the whole pile but a pilewith two (or more) less handfuls may not be. So x can be necessarily P without x being necessarily necessarily P. On the other hand, Saul Kripke has an account of de re modality which is based on (metaphysical) identity acrossworlds and is therefore transitive.Another interpretation of the 'accessibility relation' with a physical meaning was given in Gerla 1987 where the claim“is possible P in the world w′′ is interpreted as “it is possible to transform w into a world in which P is true”. So,the properties of the modal operators depend on the algebraic properties of the set of admissible transformations.There are other applications of the 'accessibility relation' in philosophy. In epistemology, one can, instead of talkingabout nomological accessibility, talk about epistemic accessibility. A world w′ is epistemically accessible from wfor an individual I in w if and only if I does not know something which would rule out the hypothesis that w′ = w. We can ask whether the relation is transitive. If I knows nothing that rules out the possibility that w′ = w andknows nothing that rules the possibility that w′′ = w′ , it does not follow that I knows nothing which rules out thehypothesis that w′′ = w . To return to our earlier example, one may not be able to distinguish a pile of sand from thesame pile with one less handful and one may not be able to distinguish the pile with one less handful from the samepile with two less handfuls of sand, but one may still be able to distinguish the original pile from the pile with twoless handfuls of sand.Yet another example of the use of the 'accessibility relation' is in deontic logic. If we think of obligatoriness as truthin all morally perfect worlds, and permissibility as truth in some morally perfect world, then we will have to restrictout universe to include only morally perfect worlds. But, in that case, we will have left out the actual world. A betteralternative would be to include all the metaphysically possible worlds but restrict the 'accessibility relation' to morallyperfect worlds. Transitivity and the euclidean property will hold, but reflexivity and symmetry will not.

1.5 Computer Science Applications

In modeling a computation, a 'possible world' can be a possible computer state. Given the current computer state, youmight define the accessible possible worlds to be all future possible computer states, or to be all possible immediate“next” computer states (assuming a discrete computer). Either choice defines a particular 'accessibility relation' givingrise to a particular modal logic suited specifically for theorems about the computation.

1.6 See also

• Modal logic

• Possible worlds

• Propositional attitude

• Modal depth

1.7 References

• Gerla, G.; Transformational semantics for first order logic, Logique et Analyse, No. 117–118, pp. 69–79,1987.

https://en.wikipedia.org/wiki/Counterpart_theory

https://en.wikipedia.org/wiki/Deontic_logic


https://en.wikipedia.org/wiki/Possible_worlds

https://en.wikipedia.org/wiki/Propositional_attitude

https://en.wikipedia.org/wiki/Modal_depth

http://www.vub.ac.be/CLWF/L&A/


• Fitelson, Brandon; Notes on “Accessibility” and Modality, 2003.

• Brown, Curtis; Propositional Modal Logic: A Few First Steps, 2002.

• Kripke, Saul; Naming and Necessity, Oxford, 1980.

• Lewis, David K.; Counterpart Theory and Quantified Modal Logic (subscription required), The Journal of Philosophy,Vol. LXV, No. 5 (1968-03-07), pp. 113–126, 1968

• List of Logic Systems List of most of the more popular modal logics.

http://www.jstor.org/sici?sici=0022-362X(19680307)65:5%253C113:CTAQML%253E2.0.CO;2-G&cookieSet=1

http://www.cc.utah.edu/~nahaj/logic/structures/systems/index.html

Chapter 2

Allegory (category theory)

In the mathematical field category theory, an allegory is a category that has some of the structure of the category ofsets and binary relations between them. Allegories can be used as an abstraction of categories of relations, and in thissense the theory of allegories is a generalization of relation algebra to relations between different sorts. Allegories arealso useful in defining and investigating certain constructions in category theory, such as exact completions.In this article we adopt the convention that morphisms compose from right to left, so RS means “first do S, then doR".

2.1 Definition

An allegory is a category in which

• every morphism R:X→Y is associated with an anti-involution, i.e. a morphism R°:Y→X; and

• every pair of morphisms R,S:X→Y with common domain/codomain is associated with an intersection, i.e. amorphism R∩S:X→Y

all such that

• intersections are idempotent (R∩R=R), commutative (R∩S=S∩R), and associative (R∩S)∩T=R∩(S∩T);

• anti-involution distributes over composition ((RS)°=S°R°) and intersection ((R∩S)°=S°∩R°);

• composition is semi-distributive over intersection (R(S∩T)⊆RS∩RT, (R∩S)T⊆RT∩ST); and

• the modularity law is satisfied: (RS∩T⊆(R∩TS°)S).

Here, we are abbreviating using the order defined by the intersection: "R⊆S" means "R=R∩S".A first example of an allegory is the category of sets and relations. The objects of this allegory are sets, and a morphismX→Y is a binary relation between X and Y. Composition of morphisms is composition of relations; intersection ofmorphisms is intersection of relations.

2.2 Regular categories and allegories

2.2.1 Allegories of relations in regular categories

In a category C, a relation between objects X, Y is a span of morphisms X←R→Y that is jointly-monic. Two suchspans X←S→Y and X←T→Y are considered equivalent when there is an isomorphism between S and T that makeeverything commute, and strictly speaking relations are only defined up to equivalence (one may formalise this eitherusing equivalence classes or using bicategories). If the category C has products, a relation between X and Y is the

7

https://en.wikipedia.org/wiki/Category_theory

https://en.wikipedia.org/wiki/Category_(mathematics)


https://en.wikipedia.org/wiki/Relation_algebra

https://en.wikipedia.org/wiki/Regular_category


https://en.wikipedia.org/wiki/Category_of_relations

https://en.wikipedia.org/wiki/Composition_of_relations

https://en.wikipedia.org/wiki/Intersection_(set_theory)

https://en.wikipedia.org/wiki/Span_(category_theory)

8 CHAPTER 2. ALLEGORY (CATEGORY THEORY)

same thing as a monomorphism into X×Y (or an equivalence class of such). In the presence of pullbacks and a properfactorization system, one can define the composition of relations. The composition of X←R→Y←S→Z is found byfirst pulling back the cospan R→Y←S and then taking the jointly-monic image of the resulting span X←R←·→S→Z.Composition of relations will be associative if the factorization system is appropriately stable. In this case one canconsider a category Rel(C), with the same objects as C, but where morphisms are relations between the objects. Theidentity relations are the diagonals X→X×X.Recall that a regular category is a category with finite limits and images in which covers are stable under pullback. Aregular category has a stable regular epi/mono factorization system. The category of relations for a regular categoryis always an allegory. Anti-involution is defined by turning the source/target of the relation around, and intersectionsare intersections of subobjects, computed by pullback.

2.2.2 Maps in allegories, and tabulations

A morphism R in an allegory A is called a map if it is entire (1⊆R°R) and deterministic (RR°⊆1). Another way ofsaying this: a map is a morphism that has a right adjoint in A, when A is considered, using the local order structure,as a 2-category. Maps in an allegory are closed under identity and composition. Thus there is a subcategory Map(A)of A, with the same objects but only the maps as morphisms. For a regular category C, there is an isomorphism ofcategories C≅Map(Rel(C)). In particular, a morphism in Map(Rel(Set)) is just an ordinary set function.In an allegory, a morphism R:X→Y is tabulated by a pair of maps f:Z→X, g:Z→Y if gf°=R and f°f∩g°g=1. Anallegory is called tabular if every morphism has a tabulation. For a regular category C, the allegory Rel(C) is alwaystabular. On the other hand, for any tabular allegory A, the category Map(A) of maps is a locally regular category: ithas pullbacks, equalizers and images that are stable under pullback. This is enough to study relations in Map(A) and,in this setting, A≅Rel(Map(A)).

2.2.3 Unital allegories and regular categories of maps

A unit in an allegory is an object U for which the identity is the largest morphism U→U, and such that from everyother object there is an entire relation to U. An allegory with a unit is called unital. Given a tabular allegory A, thecategory Map(A) is a regular category (it has a terminal object) if and only if A is unital.

2.2.4 More sophisticated kinds of allegory

Additional properties of allegories can be axiomatized. Distributive allegories have a union-like operation that issuitably well-behaved, and division allegories have a generalization of the division operation of relation algebra.Power allegories are distributive division allegories with additional powerset-like structure. The connection betweenallegories and regular categories can be developed into a connection between power allegories and toposes.

2.3 References• Peter Freyd, Andre Scedrov (1990). Categories, Allegories. Mathematical Library Vol 39. North-Holland.

ISBN 978-0-444-70368-2.

• Peter Johnstone (2003). Sketches of an Elephant: A Topos Theory Compendium. Oxford Science Publications.OUP. ISBN 0-19-852496-X.

https://en.wikipedia.org/wiki/Monomorphism

https://en.wikipedia.org/wiki/Factorization_system

https://en.wikipedia.org/wiki/Regular_category

https://en.wikipedia.org/wiki/Adjoint_functor

https://en.wikipedia.org/wiki/2-category

https://en.wikipedia.org/wiki/Union_(set_theory)


https://en.wikipedia.org/wiki/Powerset

https://en.wikipedia.org/wiki/Toposes

https://en.wikipedia.org/wiki/Peter_J._Freyd

https://en.wikipedia.org/wiki/Elsevier

https://en.wikipedia.org/wiki/International_Standard_Book_Number

https://en.wikipedia.org/wiki/Special:BookSources/978-0-444-70368-2

https://en.wikipedia.org/wiki/Peter_Johnstone_(mathematician)

https://en.wikipedia.org/wiki/OUP


https://en.wikipedia.org/wiki/Special:BookSources/0-19-852496-X

Chapter 3

Alternatization

In mathematics, more specifically in multilinear algebra, the notion of alternatization (or alternatisation in BritishEnglish) is used to pass from any map to an alternating map.An alternating map is a multilinear map (e.g., a bilinear map or a multilinear form) that is equal to zero for everytuple with two adjacent elements that are equal.

3.1 Definitions

3.1.1 Alternating bilinear map

Let S be a set, A be an abelian group, and α : S × S → A be a bilinear map. Then α is said to be an alternatingbilinear map if

∀x ∈ S, α(x, x) = 0.

3.1.2 Alternating bilinear form

An alternating bilinear form is a special case of alternating bilinear map. As bilinear forms can be defined as mapsbetween vector spaces or modules, we distinguish two cases.

Vector spacesLet V be a vector space over a field K, and α : V × V → K be a bilinear form. Then α is said to be analternating bilinear form if [1][2]

∀x ∈ V, α(x, x) = 0.

ModulesLet M be a module over a ring R, and α : M ×M → R be a bilinear form. Then α is said to be analternating bilinear form if

∀x ∈M, α(x, x) = 0.

9

https://en.wikipedia.org/wiki/Mathematics

https://en.wikipedia.org/wiki/Multilinear_algebra

https://en.wikipedia.org/wiki/British_English


https://en.wikipedia.org/wiki/Map_(mathematics)

https://en.wikipedia.org/wiki/Multilinear_map

https://en.wikipedia.org/wiki/Bilinear_map

https://en.wikipedia.org/wiki/Multilinear_form

https://en.wikipedia.org/wiki/Tuple

https://en.wikipedia.org/wiki/Element_(mathematics)

https://en.wikipedia.org/wiki/Set_(mathematics)

https://en.wikipedia.org/wiki/Abelian_group


https://en.wikipedia.org/wiki/Bilinear_form

https://en.wikipedia.org/wiki/Vector_space

https://en.wikipedia.org/wiki/Module_(mathematics)


https://en.wikipedia.org/wiki/Field_(mathematics)



https://en.wikipedia.org/wiki/Ring_(mathematics)


10 CHAPTER 3. ALTERNATIZATION

3.1.3 Alternating multilinear form

An alternatingmultilinear form generalizes the concept of alternating bilinear form to n dimensions. As multilinearforms can be defined as maps between vector spaces or modules, we distinguish two cases.

Vector spacesLet V be a vector space over a field K, and α : V × V × ...× V → K be a multilinear form. Then α issaid to be an alternating multilinear form if

∀x1, x2, ..., xn ∈ V, ∀i ∈ {1, 2, ..., n− 1}, xi = xi+1 =⇒ α(x1, x2, ..., xn) = 0.

ModulesLet M be a module over a ring R, and α : M ×M × ... ×M → R be a multilinear form. Then α issaid to be an alternating multilinear form if [3]

∀x1, x2, ..., xn ∈M, ∀i ∈ {1, 2, ..., n− 1}, xi = xi+1 =⇒ α(x1, x2, ..., xn) = 0.

3.1.4 Alternatization of a bilinear map

Let S be a set, A be an abelian group, and α : S × S → A be a bilinear map. ∀x, y ∈ S, the alternatization of themap α is the map

β : S × S → A

(x, y) 7→ α(x, y)− α(y, x).

3.2 Example

• The Lie bracket is an alternating bilinear form.

3.3 Properties

• Every alternating multilinear form is antisymmetric:[4]

∀x, y ∈ S, α(x, y) + α(y, x) = 0

• If the characteristic of the ring R is not equal to 2, then every antisymmetric multilinear form is alternating.[5]

• The alternatization of an alternating map is its double.

• The alternatization of a symmetric map is zero.

• The alternatization of a bilinear map is bilinear. Most notably, the alternatization of any cocycle is bilinear. Thisfact plays a crucial role in identifying the second cohomology group of a lattice with the group of alternatingbilinear forms on a lattice.

• There may be non-bilinear maps whose alternatization is bilinear.








https://en.wikipedia.org/wiki/Module






https://en.wikipedia.org/wiki/Lie_algebra

https://en.wikipedia.org/wiki/Antisymmetric_relation

https://en.wikipedia.org/wiki/Symmetric_map


https://en.wikipedia.org/wiki/Chain_complex

https://en.wikipedia.org/wiki/Cohomology

https://en.wikipedia.org/wiki/Lattice_(group)


3.4. SEE ALSO 11

3.4 See also• Bilinear map

• Map (mathematics)

• Multilinear algebra

• Multilinear map

3.5 Notes[1] Rotman 1995, page 235.

[2] Cohn 2003, page 298.

[3] Lang 2002, page 511.

[4] Rotman 1995, page 235.

[5] Rotman 1995, page 235.

3.6 References• Cohn, P.M. (2003). Basic Algebra: Groups, Rings and Fields. Springer. ISBN 1-85233-587-4. OCLC

248833275.

• Lang, Serge (2002). Algebra. Graduate Texts in Mathematics 211 (revised 3rd ed.). Springer. ISBN 978-0-387-95385-4. OCLC 48176673.

• Rotman, Joseph J. (1995). An Introduction to the Theory of Groups. Graduate Texts in Mathematics 148 (4thed.). Springer. ISBN 0-387-94285-8. OCLC 30028913.



https://en.wikipedia.org/wiki/Multilinear_algebra



https://en.wikipedia.org/wiki/Special:BookSources/1-85233-587-4

https://en.wikipedia.org/wiki/OCLC

https://www.worldcat.org/oclc/248833275

https://en.wikipedia.org/wiki/Graduate_Texts_in_Mathematics










Chapter 4

Ancestral relation

In mathematical logic, the ancestral relation (often shortened to ancestral) of a binary relation R is its transitiveclosure, however defined in a different way, see below.Ancestral relations make their first appearance in Frege's Begriffsschrift. Frege later employed them in his Grundge-setze as part of his definition of the finite cardinals. Hence the ancestral was a key part of his search for a logicistfoundation of arithmetic.

4.1 Definition

The numbered propositions below are taken from his Begriffsschrift and recast in contemporary notation.A property P is called R-hereditary if, whenever x is P and xRy holds, then y is also P:

(Px ∧ xRy) → Py

Frege defined b to be an R-ancestor of a, written aR*b, if b has every R-hereditary property that all objects x suchthat aRx have:

76 : ⊢ aR∗b↔ ∀F [∀x(aRx→ Fx) ∧ ∀x∀y(Fx ∧ xRy → Fy) → Fb]

The ancestral is a transitive relation:

98 : ⊢ (aR∗b ∧ bR∗c) → aR∗c

Let the notation I(R) denote that R is functional (Frege calls such relations “many-one”):

115 : ⊢ I(R) ↔ ∀x∀y∀z[(xRy ∧ xRz) → y = z]

If R is functional, then the ancestral of R is what nowadays is called connected:

133 : ⊢ (I(R) ∧ aR∗b ∧ aR∗c) → (bR∗c ∨ b = c ∨ cR∗b)

4.2 Relationship to transitive closure

The Ancestral relation R∗ is equal to the transitive closure R+ of R . Indeed, R∗ is transitive (see 98 above), R∗

contains R (indeed, if aRb then, of course, b has every R-hereditary property that all objects x such that aRx have,because b is one of them), and finally, R∗ is contained in R+ (indeed, assume aR∗b ; take the property Fx to beaR+x ; then the two premises, ∀x(aRx → Fx) and ∀x∀y(Fx ∧ xRy → Fy) , are obviously satisfied; therefore,Fb , which means aR+b , by our choice of F ). See also Boolos’s book below, page 8.

12

https://en.wikipedia.org/wiki/Mathematical_logic


https://en.wikipedia.org/wiki/Transitive_closure


https://en.wikipedia.org/wiki/Frege

https://en.wikipedia.org/wiki/Begriffsschrift

https://en.wikipedia.org/wiki/Finite_set

https://en.wikipedia.org/wiki/Cardinal_number

https://en.wikipedia.org/wiki/Logicism


https://en.wikipedia.org/wiki/Property_(philosophy)#Properties_in_mathematics

https://en.wikipedia.org/wiki/Transitive_relation

https://en.wikipedia.org/wiki/Function_(mathematics)


https://en.wikipedia.org/wiki/Connectedness


4.3. DISCUSSION 13

4.3 Discussion

Principia Mathematica made repeated use of the ancestral, as does Quine’s (1951) Mathematical Logic.However, it is worth noting that the ancestral relation cannot be defined in first-order logic. It is controversial whethersecond-order logic is really “logic” at all. Quine famously claimed that it was not, despite his reliance upon it forhis 1951 book (which largely retells Principia in abbreviated form, for which second-order logic is required to fit itstheorems).

4.4 See also• Begriffsschrift

• Gottlob Frege

• Transitive closure

4.5 References• George Boolos, 1998. Logic, Logic, and Logic. Harvard Univ. Press.

• Ivor Grattan-Guinness, 2000. In Search of Mathematical Roots. Princeton Univ. Press.

• Willard Van Orman Quine, 1951 (1940). Mathematical Logic. Harvard Univ. Press. ISBN 0-674-55451-5.

4.6 External links• Stanford Encyclopedia of Philosophy: "Frege’s Logic, Theorem, and Foundations for Arithmetic" -- by Edward

N. Zalta. Section 4.2.

https://en.wikipedia.org/wiki/Principia_Mathematica

https://en.wikipedia.org/wiki/W._V._Quine


https://en.wikipedia.org/wiki/Second-order_logic


https://en.wikipedia.org/wiki/Gottlob_Frege


https://en.wikipedia.org/wiki/George_Boolos

https://en.wikipedia.org/wiki/Ivor_Grattan-Guinness

https://en.wikipedia.org/wiki/Willard_Van_Orman_Quine

https://en.wikipedia.org/wiki/Special:BookSources/0674554515

https://en.wikipedia.org/wiki/Stanford_Encyclopedia_of_Philosophy

http://plato.stanford.edu/entries/frege-theorem/

https://en.wikipedia.org/wiki/Edward_N._Zalta

https://en.wikipedia.org/wiki/Edward_N._Zalta

Chapter 5

Antisymmetric relation

In mathematics, a binary relation R on a set X is antisymmetric if there is no pair of distinct elements of X each ofwhich is related by R to the other. More formally, R is antisymmetric precisely if for all a and b in X

if R(a,b) and R(b,a), then a = b,

or, equivalently,

if R(a,b) with a ≠ b, then R(b,a) must not hold.

As a simple example, the divisibility order on the natural numbers is an antisymmetric relation. And what antisym-metry means here is that the only way each of two numbers can be divisible by the other is if the two are, in fact, thesame number; equivalently, if n and m are distinct and n is a factor of m, then m cannot be a factor of n.In mathematical notation, this is:

∀a, b ∈ X, R(a, b) ∧R(b, a) ⇒ a = b

or, equivalently,

∀a, b ∈ X, R(a, b) ∧ a ̸= b⇒ ¬R(b, a).

The usual order relation ≤ on the real numbers is antisymmetric: if for two real numbers x and y both inequalitiesx ≤ y and y ≤ x hold then x and y must be equal. Similarly, the subset order ⊆ on the subsets of any given set isantisymmetric: given two sets A and B, if every element in A also is in B and every element in B is also in A, then Aand B must contain all the same elements and therefore be equal:

A ⊆ B ∧B ⊆ A⇒ A = B

Partial and total orders are antisymmetric by definition. A relation can be both symmetric and antisymmetric (e.g.,the equality relation), and there are relations which are neither symmetric nor antisymmetric (e.g., the “preys on”relation on biological species).Antisymmetry is different from asymmetry, which requires both antisymmetry and irreflexivity.

5.1 Examples

The relation "x is even, y is odd” between a pair (x, y) of integers is antisymmetric:

14




https://en.wikipedia.org/wiki/Divisibility

https://en.wikipedia.org/wiki/Natural_number

https://en.wikipedia.org/wiki/Mathematical_notation

https://en.wikipedia.org/wiki/Order_relation

https://en.wikipedia.org/wiki/Real_number

https://en.wikipedia.org/wiki/Inequality_(mathematics)

https://en.wikipedia.org/wiki/Subset_order


https://en.wikipedia.org/wiki/Partial_order

https://en.wikipedia.org/wiki/Total_order

https://en.wikipedia.org/wiki/Symmetric_relation

https://en.wikipedia.org/wiki/Equality_(mathematics)

https://en.wikipedia.org/wiki/Species

https://en.wikipedia.org/wiki/Asymmetric_relation

https://en.wikipedia.org/wiki/Reflexive_relation

https://en.wikipedia.org/wiki/Integer

5.2. SEE ALSO 15

Every asymmetric relation is also an antisymmetric relation.

5.2 See also• Symmetric relation

• Asymmetric relation

• Symmetry in mathematics

5.3 References• Weisstein, Eric W., “Antisymmetric Relation”, MathWorld.

• Lipschutz, Seymour; Marc Lars Lipson (1997). Theory and Problems of Discrete Mathematics. McGraw-Hill.p. 33. ISBN 0-07-038045-7.




https://en.wikipedia.org/wiki/Symmetry_in_mathematics

https://en.wikipedia.org/wiki/Eric_W._Weisstein

http://mathworld.wolfram.com/AntisymmetricRelation.html

https://en.wikipedia.org/wiki/MathWorld

https://en.wikipedia.org/wiki/Seymour_Lipschutz



Chapter 6

Asymmetric relation

In mathematics an asymmetric relation is a binary relation on a set X where:

• For all a and b in X, if a is related to b, then b is not related to a.[1]

In mathematical notation, this is:

∀a, b ∈ X, aRb ⇒ ¬(bRa)

6.1 Examples

An example is < (less-than): if x < y, then necessarily y is not less than x. In fact, one of Tarski’s axioms characterizingthe real numbers R is that < over R is asymmetric.An asymmetric relation need not be total. For example, strict subset or ⊊ is asymmetric, and neither of the sets {1,2}and {3,4} is a strict subset of the other. In general, every strict partial order is asymmetric, and conversely, everytransitive asymmetric relation is a strict partial order.Not all asymmetric relations are strict partial orders, however. An example of an asymmetric intransitive relation isthe rock-paper-scissors relation: if X beats Y, then Y does not beat X, but no one choice wins all the time.The ≤ (less than or equal) operator, on the other hand, is not asymmetric, because reversing x ≤ x produces x ≤ xand both are true. In general, any relation in which x R x holds for some x (that is, which is not irreflexive) is also notasymmetric.Asymmetric is not the same thing as “not symmetric": a relation can be neither symmetric nor asymmetric, such as≤, or can be both, only in the case of the empty relation (vacuously).

6.2 Properties• A relation is asymmetric if and only if it is both antisymmetric and irreflexive.[2]

• Restrictions and inverses of asymmetric relations are also asymmetric. For example, the restriction of < fromthe reals to the integers is still asymmetric, and the inverse > of < is also asymmetric.

• A transitive relation is asymmetric if and only if it is irreflexive:[3] if a R b and b R a, transitivity gives a R a,contradicting irreflexivity.

6.3 See also• Symmetric relation

16




https://en.wikipedia.org/wiki/Tarski%2527s_axiomatization_of_the_reals

https://en.wikipedia.org/wiki/Tarski%2527s_axiomatization_of_the_reals

https://en.wikipedia.org/wiki/Total_relation

https://en.wikipedia.org/wiki/Strict_subset

https://en.wikipedia.org/wiki/Strict_partial_order


https://en.wikipedia.org/wiki/Intransitivity

https://en.wikipedia.org/wiki/Rock-paper-scissors



https://en.wikipedia.org/wiki/Vacuous_truth



https://en.wikipedia.org/wiki/Binary_relation#Restriction

https://en.wikipedia.org/wiki/Inverse_relation



6.4. REFERENCES 17

• Antisymmetric relation

• Symmetry

• Symmetry in mathematics

6.4 References[1] Gries, David; Schneider, Fred B. (1993), A Logical Approach to Discrete Math, Springer-Verlag, p. 273.

[2] Nievergelt, Yves (2002), Foundations of Logic and Mathematics: Applications to Computer Science and Cryptography,Springer-Verlag, p. 158.

[3] Flaška, V.; Ježek, J.; Kepka, T.; Kortelainen, J. (2007). Transitive Closures of Binary Relations I (PDF). Prague: Schoolof Mathematics - Physics Charles University. p. 1. Lemma 1.1 (iv). Note that this source refers to asymmetric relationsas “strictly antisymmetric”.


https://en.wikipedia.org/wiki/Symmetry


https://en.wikipedia.org/wiki/David_Gries

https://en.wikipedia.org/wiki/Fred_B._Schneider

http://books.google.com/books?id=ZWTDQ6H6gsUC&pg=PA273

http://books.google.com/books?id=_H_nJdagqL8C&pg=PA158

http://www.karlin.mff.cuni.cz/~jezek/120/transitive1.pdf

Chapter 7

Better-quasi-ordering

In order theory a better-quasi-ordering or bqo is a quasi-ordering that does not admit a certain type of bad array.Every bqo is well-quasi-ordered.

7.1 Motivation

Though wqo is an appealing notion, many important infinitary operations do not preserve wqoness. An exampledue to Richard Rado illustrates this.[1] In a 1965 paper Crispin Nash-Williams formulated the stronger notion ofbqo in order to prove that the class of trees of height ω is wqo under the topological minor relation.[2] Since then,many quasi-orders have been proven to be wqo by proving them to be bqo. For instance, Richard Laver establishedFraïssé's conjecture by proving that the class of scattered linear order types is bqo.[3] More recently, Carlos Martinez-Ranero has proven that, under the Proper Forcing Axiom, the class of Aronszajn lines is bqo under the embeddabilityrelation.[4]

7.2 Definition

It is common in bqo theory to write ∗x for the sequence x with the first term omitted. Write [ω]<ω for the set offinite, strictly increasing sequences with terms in ω , and define a relation ◁ on [ω]<ω as follows: s ◁ t if and only ifthere is u such that s is a strict initial segment of u and t = ∗u . Note that the relation ◁ is not transitive.A block is a subset B of [ω]<ω that contains an initial segment of every infinite subset of

∪B . For a quasi-order Q

a Q -pattern is a function from a block B into Q . A Q -pattern f : B → Q is said to be bad if f(s) ̸≤Q f(t) forevery pair s, t ∈ B such that s ◁ t ; otherwise f is good. A quasi-order Q is better-quasi-ordered (bqo) if there is nobad Q -pattern.In order to make this definition easier to work with, Nash-Williams defines a barrier to be a block whose elementsare pairwise incomparable under the inclusion relation ⊂ . A Q -array is aQ -pattern whose domain is a barrier. Byobserving that every block contains a barrier, one sees that Q is bqo if and only if there is no bad Q -array.

7.3 Simpson’s alternative definition

Simpson introduced an alternative definition of bqo in terms of Borel maps [ω]ω → Q , where [ω]ω , the set of infinitesubsets of ω , is given the usual (product) topology.[5]

Let Q be a quasi-order and endow Q with the discrete topology. A Q -array is a Borel function [A]ω → Q forsome infinite subset A of ω . A Q -array f is bad if f(X) ̸≤Q f(∗X) for every X ∈ [A]ω ; f is good otherwise.The quasi-order Q is bqo if there is no bad Q -array in this sense.

18

https://en.wikipedia.org/wiki/Order_theory

https://en.wikipedia.org/wiki/Quasi-ordering

https://en.wikipedia.org/wiki/Well-quasi-order

https://en.wikipedia.org/wiki/Wqo


https://en.wikipedia.org/wiki/Richard_Rado

https://en.wikipedia.org/wiki/Nash-Williams

https://en.wikipedia.org/wiki/Tree_(set_theory)


https://en.wikipedia.org/wiki/Topological_minor

https://en.wikipedia.org/wiki/Quasi-order


https://en.wikipedia.org/wiki/Richard_Laver

https://en.wikipedia.org/wiki/Roland_Fra%C3%AFss%C3%A9

https://en.wikipedia.org/wiki/Linear_order

https://en.wikipedia.org/wiki/Proper_Forcing_Axiom

https://en.wikipedia.org/wiki/Aronszajn_line




https://en.wikipedia.org/wiki/Comparability

https://en.wikipedia.org/wiki/Discrete_topology

https://en.wikipedia.org/wiki/Borel_function

7.4. MAJOR THEOREMS 19

7.4 Major theorems

Many major results in bqo theory are consequences of the Minimal Bad Array Lemma, which appears in Simpson’spaper[5] as follows. See also Laver’s paper,[6] where the Minimal Bad Array Lemma was first stated as a result. Thetechnique was present in Nash-Williams’ original 1965 paper.Suppose (Q,≤Q) is a quasi-order. A partial ranking ≤′ of Q is a well-founded partial ordering of Q such thatq ≤′ r → q ≤Q r . For bad Q -arrays (in the sense of Simpson) f : [A]ω → Q and g : [B]ω → Q , define:

g ≤∗ f if B ⊆ A and g(X) ≤′ f(X) every for X ∈ [B]ω

g <∗ f if B ⊆ A and g(X) <′ f(X) every for X ∈ [B]ω

We say a bad Q -array g is minimal bad (with respect to the partial ranking ≤′ ) if there is no bad Q -array f suchthat f <∗ g . Note that the definitions of ≤∗ and<′ depend on a partial ranking ≤′ ofQ . Note also that the relation<∗ is not the strict part of the relation ≤∗ .Theorem (Minimal Bad Array Lemma). Let Q be a quasi-order equipped with a partial ranking and suppose f is abad Q -array. Then there is a minimal bad Q -array g such that g ≤∗ f .

7.5 See also• Well-quasi-ordering

• Well-order

7.6 References[1] Rado, Richard (1954). “Partial well-ordering of sets of vectors”. Mathematika 1 (2): 89–95. doi:10.1112/S0025579300000565.

MR 0066441.

[2] Nash-Williams, C. St. J. A. (1965). “On well-quasi-ordering infinite trees”. Mathematical Proceedings of the CambridgePhilosophical Society 61 (3): 697–720. Bibcode:1965PCPS...61..697N. doi:10.1017/S0305004100039062. ISSN 0305-0041. MR 0175814.

[3] Laver, Richard (1971). “On Fraisse’s Order Type Conjecture”. TheAnnals ofMathematics 93 (1): 89–111. doi:10.2307/1970754.JSTOR 1970754.

[4] Martinez-Ranero, Carlos (2011). “Well-quasi-ordering Aronszajn lines”. Fundamenta Mathematicae 213 (3): 197–211.doi:10.4064/fm213-3-1. ISSN 0016-2736. MR 2822417.

[5] Simpson, Stephen G. (1985). “BQO Theory and Fraïssé's Conjecture”. In Mansfield, Richard; Weitkamp, Galen. RecursiveAspects of Descriptive Set Theory. The Clarendon Press, Oxford University Press. pp. 124–38. ISBN 978-0-19-503602-2.MR 786122.

[6] Laver, Richard (1978). “Better-quasi-orderings and a class of trees”. In Rota, Gian-Carlo. Studies in foundations andcombinatorics. Academic Press. pp. 31–48. ISBN 978-0-12-599101-8. MR 0520553.


https://en.wikipedia.org/wiki/Well-founded



https://en.wikipedia.org/wiki/Well-quasi-ordering

https://en.wikipedia.org/wiki/Well-order

https://en.wikipedia.org/wiki/Richard_Rado

https://en.wikipedia.org/wiki/Digital_object_identifier

https://dx.doi.org/10.1112%252FS0025579300000565

https://en.wikipedia.org/wiki/Mathematical_Reviews

https://www.ams.org/mathscinet-getitem?mr=0066441

https://en.wikipedia.org/wiki/Crispin_Nash-Williams

https://en.wikipedia.org/wiki/Bibcode

http://adsabs.harvard.edu/abs/1965PCPS...61..697N



https://en.wikipedia.org/wiki/International_Standard_Serial_Number

https://www.worldcat.org/issn/0305-0041





https://dx.doi.org/10.2307%252F1970754

https://en.wikipedia.org/wiki/JSTOR

https://www.jstor.org/stable/1970754


https://dx.doi.org/10.4064%252Ffm213-3-1

https://en.wikipedia.org/wiki/International_Standard_Serial_Number












Chapter 8

Bidirectional transformation

In computer programming, bidirectional transformations (bx) are programs in which a single piece of code canbe run in several ways, such that the same data are sometimes considered as input, and sometimes as output. Forexample, a bx run in the forward direction might transform input I into output O, while the same bx run backwardwould take as input versions of I and O and produce a new version of I as its output.Bidirectional model transformations are an important special case in which a model is input to such a program.Some bidirectional languages are bijective. The bijectivity of a language is a severe restriction of its bidirectionality,[1]

because a bijective language is merely relating two different ways to present the very same information.More general is a lens language, in which there is a distinguished forward direction (“get”) that takes a concrete inputto an abstract output, discarding some information in the process: the concrete state includes all the information thatis in the abstract state, and usually some more. The backward direction (“put”) takes a concrete state and an abstractstate and computes a new concrete state. Lenses are required to obey certain conditions to ensure sensible behaviour.The most general case is that of symmetric bidirectional transformations. Here the two states that are related typicallyshare some information, but each also includes some information that is not included in the other.

8.1 Usage

Bidirectional transformations can be used to:

• Maintain several sources of information consistent[2]

• Provide an 'abstract view' to easily manipulate data and write them back to their source

8.2 Vocabulary

A bidirectional program which obeys certain round-trip laws is called a lens.

8.3 Examples of implementations

• Boomerang is a programming language which allows to write lenses to process text data formats bidirectionally

• Augeas is a configuration management library whose lens language is inspired by the Boomerang project

• biXid is a programming language to process XML data bidirectionally[3]

• XSugar allows to translate from XML to non-XML formats[4]

20

https://en.wikipedia.org/wiki/Model_transformation#Unidirectional_versus_bidirectional

https://en.wikipedia.org/wiki/Bijection

https://en.wikipedia.org/wiki/Boomerang_(programming_language)

https://en.wikipedia.org/wiki/Augeas_(software)

https://en.wikipedia.org/wiki/Bixid_(programming_language)

https://en.wikipedia.org/wiki/XSugar_(programming_language)

8.4. SEE ALSO 21

8.4 See also• Bidirectionalization

8.5 References[1] http://grace.gsdlab.org/images/e/e2/Nate-short.pdf

[2] http://www.cs.cornell.edu/~{}jnfoster/papers/grace-report.pdf

[3] http://arbre.is.s.u-tokyo.ac.jp/~{}hahosoya/papers/bixid.pdf

[4] http://www.brics.dk/xsugar/

8.6 External links• GRACE International Meeting on Bidirectional Transformations

• Bidirectional Transformations: The Bx Wiki

https://en.wikipedia.org/wiki/Bidirectionalization

http://grace.gsdlab.org/images/e/e2/Nate-short.pdf

http://www.cs.cornell.edu/~jnfoster/papers/grace-report.pdf

http://arbre.is.s.u-tokyo.ac.jp/~hahosoya/papers/bixid.pdf

http://www.brics.dk/xsugar/

http://grace.gsdlab.org/

http://bx-community.wikidot.com/

Chapter 9

Bijection

X 1

2

3

4

YD

B

C

A

A bijective function, f: X → Y, where set X is {1, 2, 3, 4} and set Y is {A, B, C, D}. For example, f(1) = D.

In mathematics, a bijection, bijective function or one-to-one correspondence is a function between the elementsof two sets, where every element of one set is paired with exactly one element of the other set, and every elementof the other set is paired with exactly one element of the first set. There are no unpaired elements. In mathematical

22



9.1. DEFINITION 23

terms, a bijective function f: X → Y is a one-to-one (injective) and onto (surjective) mapping of a set X to a set Y.A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existenceof a bijection means they have the same number of elements. For infinite sets the picture is more complicated, leadingto the concept of cardinal number, a way to distinguish the various sizes of infinite sets.A bijective function from a set to itself is also called a permutation.Bijective functions are essential to many areas of mathematics including the definitions of isomorphism, homeomorphism,diffeomorphism, permutation group, and projective map.

9.1 Definition

For more details on notation, see Function (mathematics) § Notation.

For a pairing between X and Y (where Y need not be different from X) to be a bijection, four properties must hold:

1. each element of X must be paired with at least one element of Y,

2. no element of X may be paired with more than one element of Y,

3. each element of Y must be paired with at least one element of X, and

4. no element of Y may be paired with more than one element of X.

Satisfying properties (1) and (2) means that a bijection is a function with domain X. It is more common to seeproperties (1) and (2) written as a single statement: Every element of X is paired with exactly one element of Y.Functions which satisfy property (3) are said to be "onto Y " and are called surjections (or surjective functions).Functions which satisfy property (4) are said to be "one-to-one functions" and are called injections (or injectivefunctions).[1] With this terminology, a bijection is a function which is both a surjection and an injection, or usingother words, a bijection is a function which is both “one-to-one” and “onto”.

9.2 Examples

9.2.1 Batting line-up of a baseball team

Consider the batting line-up of a baseball team (or any list of all the players of any sports team). The set X will be thenine players on the team and the set Y will be the nine positions in the batting order (1st, 2nd, 3rd, etc.) The “pairing”is given by which player is in what position in this order. Property (1) is satisfied since each player is somewhere inthe list. Property (2) is satisfied since no player bats in two (or more) positions in the order. Property (3) says thatfor each position in the order, there is some player batting in that position and property (4) states that two or moreplayers are never batting in the same position in the list.

9.2.2 Seats and students of a classroom

In a classroom there are a certain number of seats. A bunch of students enter the room and the instructor asks themall to be seated. After a quick look around the room, the instructor declares that there is a bijection between the setof students and the set of seats, where each student is paired with the seat they are sitting in. What the instructorobserved in order to reach this conclusion was that:

1. Every student was in a seat (there was no one standing),

2. No student was in more than one seat,

3. Every seat had someone sitting there (there were no empty seats), and

4. No seat had more than one student in it.

https://en.wikipedia.org/wiki/Injective_function

https://en.wikipedia.org/wiki/Surjective_function

https://en.wikipedia.org/wiki/Inverse_function


https://en.wikipedia.org/wiki/Infinite_set


https://en.wikipedia.org/wiki/Permutation

https://en.wikipedia.org/wiki/Isomorphism

https://en.wikipedia.org/wiki/Homeomorphism

https://en.wikipedia.org/wiki/Diffeomorphism

https://en.wikipedia.org/wiki/Permutation_group

https://en.wikipedia.org/wiki/Projective_map

https://en.wikipedia.org/wiki/Function_(mathematics)#Notation


https://en.wikipedia.org/wiki/Domain_of_a_function

https://en.wikipedia.org/wiki/Onto


https://en.wikipedia.org/wiki/One-to-one_function


https://en.wikipedia.org/wiki/Batting_order_(baseball)

https://en.wikipedia.org/wiki/Baseball

24 CHAPTER 9. BIJECTION

The instructor was able to conclude that there were just as many seats as there were students, without having to counteither set.

9.3 More mathematical examples and some non-examples• For any set X, the identity function 1X: X → X, 1X(x) = x, is bijective.

• The function f: R → R, f(x) = 2x + 1 is bijective, since for each y there is a unique x = (y − 1)/2 such that f(x)= y. In more generality, any linear function over the reals, f: R → R, f(x) = ax + b (where a is non-zero) is abijection. Each real number y is obtained from (paired with) the real number x = (y - b)/a.

• The function f: R → (-π/2, π/2), given by f(x) = arctan(x) is bijective since each real number x is pairedwith exactly one angle y in the interval (-π/2, π/2) so that tan(y) = x (that is, y = arctan(x)). If the codomain(-π/2, π/2) was made larger to include an integer multiple of π/2 then this function would no longer be onto(surjective) since there is no real number which could be paired with the multiple of π/2 by this arctan function.

• The exponential function, g: R → R, g(x) = ex, is not bijective: for instance, there is no x in R such that g(x) =−1, showing that g is not onto (surjective). However if the codomain is restricted to the positive real numbersR+ ≡ (0,+∞) , then g becomes bijective; its inverse (see below) is the natural logarithm function ln.

• The function h: R → R+, h(x) = x2 is not bijective: for instance, h(−1) = h(1) = 1, showing that h is not one-to-one (injective). However, if the domain is restricted to R+

0 ≡ [0,+∞) , then h becomes bijective; its inverseis the positive square root function.

9.4 Inverses

A bijection f with domain X (“functionally” indicated by f: X→Y) also defines a relation starting in Y and going to X(by turning the arrows around). The process of “turning the arrows around” for an arbitrary function does not usuallyyield a function, but properties (3) and (4) of a bijection say that this inverse relation is a function with domain Y.Moreover, properties (1) and (2) then say that this inverse function is a surjection and an injection, that is, the inversefunction exists and is also a bijection. Functions that have inverse functions are said to be invertible. A function isinvertible if and only if it is a bijection.Stated in concise mathematical notation, a function f: X → Y is bijective if and only if it satisfies the condition

for every y in Y there is a unique x in X with y = f(x).

Continuing with the baseball batting line-up example, the function that is being defined takes as input the name ofone of the players and outputs the position of that player in the batting order. Since this function is a bijection, it hasan inverse function which takes as input a position in the batting order and outputs the player who will be batting inthat position.

9.5 Composition

The composition g ◦ f of two bijections f: X → Y and g: Y → Z is a bijection. The inverse of g ◦ f is (g ◦ f)−1 =

(f−1) ◦ (g−1) .Conversely, if the composition g ◦ f of two functions is bijective, we can only say that f is injective and g is surjective.

9.6 Bijections and cardinality

If X and Y are finite sets, then there exists a bijection between the two sets X and Y if and only if X and Y havethe same number of elements. Indeed, in axiomatic set theory, this is taken as the definition of “same number ofelements” (equinumerosity), and generalising this definition to infinite sets leads to the concept of cardinal number,a way to distinguish the various sizes of infinite sets.

https://en.wikipedia.org/wiki/Identity_function

https://en.wikipedia.org/wiki/Linear_function

https://en.wikipedia.org/wiki/Codomain

https://en.wikipedia.org/wiki/Exponential_function

https://en.wikipedia.org/wiki/Natural_logarithm


https://en.wikipedia.org/wiki/Relation_(mathematics)




https://en.wikipedia.org/wiki/Invertible_function

https://en.wikipedia.org/wiki/Composition_(mathematics)




https://en.wikipedia.org/wiki/Iff

https://en.wikipedia.org/wiki/Axiomatic_set_theory

https://en.wikipedia.org/wiki/Equinumerosity

https://en.wikipedia.org/wiki/Infinite_set


9.7. PROPERTIES 25

X1

2

3

YD

B

C

A

ZP

Q

R

A bijection composed of an injection (left) and a surjection (right).

9.7 Properties

• A function f: R → R is bijective if and only if its graph meets every horizontal and vertical line exactly once.

• If X is a set, then the bijective functions from X to itself, together with the operation of functional composition(∘), form a group, the symmetric group of X, which is denoted variously by S(X), SX, or X! (X factorial).

• Bijections preserve cardinalities of sets: for a subset A of the domain with cardinality |A| and subset B of thecodomain with cardinality |B|, one has the following equalities:

|f(A)| = |A| and |f−1(B)| = |B|.

• If X and Y are finite sets with the same cardinality, and f: X → Y, then the following are equivalent:

1. f is a bijection.2. f is a surjection.3. f is an injection.

• For a finite set S, there is a bijection between the set of possible total orderings of the elements and the set ofbijections from S to S. That is to say, the number of permutations of elements of S is the same as the numberof total orderings of that set—namely, n!.

9.8 Bijections and category theory

Bijections are precisely the isomorphisms in the category Set of sets and set functions. However, the bijections are notalways the isomorphisms for more complex categories. For example, in the category Grp of groups, the morphismsmust be homomorphisms since they must preserve the group structure, so the isomorphisms are group isomorphismswhich are bijective homomorphisms.

https://en.wikipedia.org/wiki/Graph_of_a_function

https://en.wikipedia.org/wiki/Group_(algebra)

https://en.wikipedia.org/wiki/Symmetric_group

https://en.wikipedia.org/wiki/Factorial

https://en.wikipedia.org/wiki/Cardinalities


https://en.wikipedia.org/wiki/Surjection

https://en.wikipedia.org/wiki/Injection_(mathematics)

https://en.wikipedia.org/wiki/Total_ordering




https://en.wikipedia.org/wiki/Category_of_sets


https://en.wikipedia.org/wiki/Category_of_groups

https://en.wikipedia.org/wiki/Group_(mathematics)

https://en.wikipedia.org/wiki/Homomorphism

26 CHAPTER 9. BIJECTION

9.9 Generalization to partial functions

The notion of one-one correspondence generalizes to partial functions, where they are called partial bijections,although partial bijections are only required to be injective. The reason for this relaxation is that a (proper) partialfunction is already undefined for a portion of its domain; thus there is no compelling reason to constrain its inverseto be a total function, i.e. defined everywhere on its domain. The set of all partial bijections on a given base set iscalled the symmetric inverse semigroup.[2]

Another way of defining the same notion is to say that a partial bijection from A to B is any relation R (which turnsout to be a partial function) with the property that R is the graph of a bijection f:A′→B′, where A′ is a subset of Aand likewise B′⊆B.[3]

When the partial bijection is on the same set, it is sometimes called a one-to-one partial transformation.[4] Anexample is the Möbius transformation simply defined on the complex plane, rather than its completion to the extendedcomplex plane.[5]

9.10 Contrast withThis list is incomplete; you can help by expanding it.

• Multivalued function

9.11 See also• Injective function

• Surjective function

• Bijection, injection and surjection

• Symmetric group

• Bijective numeration

• Bijective proof

• Cardinality

• Category theory

• Ax–Grothendieck theorem

9.12 Notes[1] There are names associated to properties (1) and (2) as well. A relation which satisfies property (1) is called a total relation

and a relation satisfying (2) is a single valued relation.

[2] Christopher Hollings (16 July 2014). Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups.American Mathematical Society. p. 251. ISBN 978-1-4704-1493-1.

[3] Francis Borceux (1994). Handbook of Categorical Algebra: Volume 2, Categories and Structures. Cambridge UniversityPress. p. 289. ISBN 978-0-521-44179-7.

[4] Pierre A. Grillet (1995). Semigroups: An Introduction to the Structure Theory. CRC Press. p. 228. ISBN 978-0-8247-9662-4.

[5] John Meakin (2007). “Groups and semigroups: connections and contrasts”. In C.M. Campbell, M.R. Quick, E.F. Robert-son, G.C. Smith. Groups St Andrews 2005 Volume 2. Cambridge University Press. p. 367. ISBN 978-0-521-69470-4.preprint citing Lawson, M. V. (1998). “The Möbius Inverse Monoid”. Journal of Algebra 200 (2): 428. doi:10.1006/jabr.1997.7242.

https://en.wikipedia.org/wiki/Partial_functions

https://en.wikipedia.org/wiki/Total_function

https://en.wikipedia.org/wiki/Symmetric_inverse_semigroup


https://en.wikipedia.org/wiki/Subset

https://en.wikipedia.org/wiki/Transformation_(function)

https://en.wikipedia.org/wiki/M%C3%B6bius_transformation

https://en.wikipedia.org/wiki/Wikipedia:WikiProject_Lists#Incomplete_lists

https://en.wikipedia.org/w/index.php?title=Bijection&action=edit

https://en.wikipedia.org/wiki/Multivalued_function



https://en.wikipedia.org/wiki/Bijection,_injection_and_surjection


https://en.wikipedia.org/wiki/Bijective_numeration

https://en.wikipedia.org/wiki/Bijective_proof

https://en.wikipedia.org/wiki/Cardinality


https://en.wikipedia.org/wiki/Ax%E2%80%93Grothendieck_theorem

http://books.google.com/books?id=O9wJBAAAQBAJ&pg=PA251



http://books.google.com/books?id=5i2v9q0m5XAC&pg=PA289



http://books.google.com/books?id=yM544W1N2UUC&pg=PA228






http://www.math.unl.edu/~jmeakin2/groups%2520and%2520semigroups.pdf


https://dx.doi.org/10.1006%252Fjabr.1997.7242

9.13. REFERENCES 27

9.13 References

This topic is a basic concept in set theory and can be found in any text which includes an introduction to set theory.Almost all texts that deal with an introduction to writing proofs will include a section on set theory, so the topic maybe found in any of these:

• Wolf (1998). Proof, Logic and Conjecture: A Mathematician’s Toolbox. Freeman.

• Sundstrom (2003). Mathematical Reasoning: Writing and Proof. Prentice-Hall.

• Smith; Eggen; St.Andre (2006). A Transition to Advanced Mathematics (6th Ed.). Thomson (Brooks/Cole).

• Schumacher (1996). Chapter Zero: Fundamental Notions of Abstract Mathematics. Addison-Wesley.

• O'Leary (2003). The Structure of Proof: With Logic and Set Theory. Prentice-Hall.

• Morash. Bridge to Abstract Mathematics. Random House.

• Maddox (2002). Mathematical Thinking and Writing. Harcourt/ Academic Press.

• Lay (2001). Analysis with an introduction to proof. Prentice Hall.

• Gilbert; Vanstone (2005). An Introduction to Mathematical Thinking. Pearson Prentice-Hall.

• Fletcher; Patty. Foundations of Higher Mathematics. PWS-Kent.

• Iglewicz; Stoyle. An Introduction to Mathematical Reasoning. MacMillan.

• Devlin, Keith (2004). Sets, Functions, and Logic: An Introduction to Abstract Mathematics. Chapman & Hall/CRC Press.

• D'Angelo; West (2000). Mathematical Thinking: Problem Solving and Proofs. Prentice Hall.

• Cupillari. The Nuts and Bolts of Proofs. Wadsworth.

• Bond. Introduction to Abstract Mathematics. Brooks/Cole.

• Barnier; Feldman (2000). Introduction to Advanced Mathematics. Prentice Hall.

• Ash. A Primer of Abstract Mathematics. MAA.

9.14 External links• Hazewinkel, Michiel, ed. (2001), “Bijection”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-

010-4

• Weisstein, Eric W., “Bijection”, MathWorld.

• Earliest Uses of Some of the Words of Mathematics: entry on Injection, Surjection and Bijection has the historyof Injection and related terms.

http://www.encyclopediaofmath.org/index.php?title=p/b016230

https://en.wikipedia.org/wiki/Encyclopedia_of_Mathematics

https://en.wikipedia.org/wiki/Springer_Science+Business_Media





http://mathworld.wolfram.com/Bijection.html


http://jeff560.tripod.com/i.html


Chapter 10

Bijection, injection and surjection

In mathematics, injections, surjections and bijections are classes of functions distinguished by the manner in whicharguments (input expressions from the domain) and images (output expressions from the codomain) are related ormapped to each other.A function maps elements from its domain to elements in its codomain. Given a function f : A→ B

• The function is injective (one-to-one) if every element of the codomain is mapped to by at most one elementof the domain. An injective function is an injection. Notationally:

∀x, y ∈ A, f(x) = f(y) ⇒ x = y.

Or, equivalently (using logical transposition),∀x, y ∈ A, x ̸= y ⇒ f(x) ̸= f(y).

• The function is surjective (onto) if every element of the codomain is mapped to by at least one element of thedomain. (That is, the image and the codomain of the function are equal.) A surjective function is a surjection.Notationally:

∀y ∈ B, ∃x ∈ A that such y = f(x).

• The function is bijective (one-to-one and onto or one-to-one correspondence) if every element of thecodomain is mapped to by exactly one element of the domain. (That is, the function is both injective andsurjective.) A bijective function is a bijection.

An injective function need not be surjective (not all elements of the codomain may be associated with arguments),and a surjective function need not be injective (some images may be associated with more than one argument). Thefour possible combinations of injective and surjective features are illustrated in the diagrams to the right.

10.1 Injection

Main article: Injective functionFor more details on notation, see Function (mathematics) § Notation.A function is injective (one-to-one) if every possible element of the codomain is mapped to by at most one argument.

Equivalently, a function is injective if it maps distinct arguments to distinct images. An injective function is aninjection. The formal definition is the following.

The function f : A→ B is injective iff for all a, b ∈ A , we have f(a) = f(b) ⇒ a = b.

• A function f : A → B is injective if and only if A is empty or f is left-invertible; that is, there is a function g :f(A) → A such that g o f = identity function on A. Here f(A) is the image of f.

28



https://en.wikipedia.org/wiki/Parameter

https://en.wikipedia.org/wiki/Expression_(mathematics)

https://en.wikipedia.org/wiki/Domain_(mathematics)

https://en.wikipedia.org/wiki/Image_(mathematics)




https://en.wikipedia.org/wiki/Transposition_(logic)


https://en.wikipedia.org/wiki/Bijective_function




10.2. SURJECTION 29

X1

2

3

YD

B

C

A

ZP

Q

R

S

Injective composition: the second function need not be injective.

• Since every function is surjective when its codomain is restricted to its image, every injection induces a bijectiononto its image. More precisely, every injection f : A → B can be factored as a bijection followed by an inclusionas follows. Let fR : A → f(A) be f with codomain restricted to its image, and let i : f(A) → B be the inclusionmap from f(A) into B. Then f = i o fR. A dual factorisation is given for surjections below.

• The composition of two injections is again an injection, but if g o f is injective, then it can only be concludedthat f is injective. See the figure at right.

• Every embedding is injective.

10.2 Surjection

Main article: Surjective functionA function is surjective (onto) if every possible image is mapped to by at least one argument. In other words, everyelement in the codomain has non-empty preimage. Equivalently, a function is surjective if its image is equal to itscodomain. A surjective function is a surjection. The formal definition is the following.

The function f : A→ B is surjective iff for all b ∈ B , there is a ∈ A such that f(a) = b.

• A function f : A → B is surjective if and only if it is right-invertible, that is, if and only if there is a function g:B → A such that f o g = identity function on B. (This statement is equivalent to the axiom of choice.)

• By collapsing all arguments mapping to a given fixed image, every surjection induces a bijection defined on aquotient of its domain. More precisely, every surjection f : A → B can be factored as a non-bijection followedby a bijection as follows. Let A/~ be the equivalence classes of A under the following equivalence relation: x ~y if and only if f(x) = f(y). Equivalently, A/~ is the set of all preimages under f. Let P(~) : A → A/~ be theprojection map which sends each x in A to its equivalence class [x]~, and let fP : A/~ → B be the well-definedfunction given by fP([x]~) = f(x). Then f = fP o P(~). A dual factorisation is given for injections above.

• The composition of two surjections is again a surjection, but if g o f is surjective, then it can only be concludedthat g is surjective. See the figure.



https://en.wikipedia.org/wiki/Embedding


https://en.wikipedia.org/wiki/Preimage


https://en.wikipedia.org/wiki/Axiom_of_choice


30 CHAPTER 10. BIJECTION, INJECTION AND SURJECTION

X1

2

3

4

YD

B

C

A

ZP

Q

R

Surjective composition: the first function need not be surjective.

10.3 Bijection

Main article: Bijective functionA function is bijective if it is both injective and surjective. A bijective function is a bijection (one-to-one corre-spondence). A function is bijective if and only if every possible image is mapped to by exactly one argument. Thisequivalent condition is formally expressed as follow.

The function f : A→ B is bijective iff for all b ∈ B , there is a unique a ∈ A such that f(a) = b.

• A function f : A → B is bijective if and only if it is invertible, that is, there is a function g: B → A such thatg o f = identity function on A and f o g = identity function on B. This function maps each image to its uniquepreimage.

• The composition of two bijections is again a bijection, but if g o f is a bijection, then it can only be concludedthat f is injective and g is surjective. (See the figure at right and the remarks above regarding injections andsurjections.)

• The bijections from a set to itself form a group under composition, called the symmetric group.

10.4 Cardinality

Suppose you want to define what it means for two sets to “have the same number of elements”. One way to do this isto say that two sets “have the same number of elements” if and only if all the elements of one set can be paired withthe elements of the other, in such a way that each element is paired with exactly one element. Accordingly, we candefine two sets to “have the same number of elements” if there is a bijection between them. We say that the two setshave the same cardinality.Likewise, we can say that set A “has fewer than or the same number of elements” as set B if there is an injectionfrom A to B . We can also say that set A “has fewer than the number of elements” in set B if there is an injectionfrom A to B but not a bijection between A and B .


https://en.wikipedia.org/wiki/If_and_only_if





10.5. EXAMPLES 31

X1

2

3

YD

B

C

A

ZP

Q

R

Bijective composition: the first function need not be surjective and the second function need not be injective.

10.5 Examples

It is important to specify the domain and codomain of each function since by changing these, functions which wethink of as the same may have different jectivity.

10.5.1 Injective and surjective (bijective)

• For every set A the identity function idA and thus specifically R → R : x 7→ x .

• R+ → R+ : x 7→ x2 and thus also its inverse R+ → R+ : x 7→√x .

• The exponential function exp : R → R+ : x 7→ ex and thus also its inverse the natural logarithm ln : R+ →R : x 7→ lnx

10.5.2 Injective and non-surjective

• The exponential function exp : R → R : x 7→ ex

10.5.3 Non-injective and surjective

• R → R : x 7→ (x− 1)x(x+ 1) = x3 − x

• R → [−1, 1] : x 7→ sin(x)

10.5.4 Non-injective and non-surjective

• R → R : x 7→ sin(x)



32 CHAPTER 10. BIJECTION, INJECTION AND SURJECTION

10.6 Properties• For every function f, subsetA of the domain and subsetB of the codomain we haveA⊂ f −1(f(A)) and f(f −1(B))

⊂ B. If f is injective we have A = f −1(f(A)) and if f is surjective we have f(f −1(B)) = B.

• For every function h : A → C we can define a surjection H : A → h(A) : a → h(a) and an injection I : h(A) →C : a → a. It follows that h = I ∘ H. This decomposition is unique up to isomorphism.

10.7 Category theory

In the category of sets, injections, surjections, and bijections correspond precisely to monomorphisms, epimorphisms,and isomorphisms, respectively.

10.8 History

This terminology was originally coined by the Bourbaki group.

10.9 See also• Bijective function

• Horizontal line test

• Injective module

• Injective function

• Permutation


10.10 External links• Earliest Uses of Some of the Words of Mathematics: entry on Injection, Surjection and Bijection has the history

of Injection and related terms.

https://en.wikipedia.org/wiki/Up_to_isomorphism



https://en.wikipedia.org/wiki/Monomorphism

https://en.wikipedia.org/wiki/Epimorphism


https://en.wikipedia.org/wiki/Bourbaki


https://en.wikipedia.org/wiki/Horizontal_line_test

https://en.wikipedia.org/wiki/Injective_module






Chapter 11

Homogeneous relation

“Relation (mathematics)" redirects here. For a more general notion of relation, see finitary relation. For a morecombinatorial viewpoint, see theory of relations. For other uses, see Relation § Mathematics.

In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is asubset of the Cartesian product A2 = A × A. More generally, a binary relation between two sets A and B is a subsetof A × B. The terms correspondence, dyadic relation and 2-place relation are synonyms for binary relation.An example is the "divides" relation between the set of prime numbers P and the set of integers Z, in which everyprime p is associated with every integer z that is a multiple of p (but with no integer that is not a multiple of p). Inthis relation, for instance, the prime 2 is associated with numbers that include −4, 0, 6, 10, but not 1 or 9; and theprime 3 is associated with numbers that include 0, 6, and 9, but not 4 or 13.Binary relations are used in many branches of mathematics to model concepts like "is greater than", "is equal to", and“divides” in arithmetic, "is congruent to" in geometry, “is adjacent to” in graph theory, “is orthogonal to” in linearalgebra and many more. The concept of function is defined as a special kind of binary relation. Binary relations arealso heavily used in computer science.A binary relation is the special case n = 2 of an n-ary relation R ⊆ A1 × … × An, that is, a set of n-tuples where thejth component of each n-tuple is taken from the jth domain Aj of the relation. An example for a ternary relation onZ×Z×Z is “lies between ... and ...”, containing e.g. the triples (5,2,8), (5,8,2), and (−4,9,−7).In some systems of axiomatic set theory, relations are extended to classes, which are generalizations of sets. Thisextension is needed for, among other things, modeling the concepts of “is an element of” or “is a subset of” in settheory, without running into logical inconsistencies such as Russell’s paradox.

11.1 Formal definition

A binary relation R is usually defined as an ordered triple (X, Y, G) where X and Y are arbitrary sets (or classes), andG is a subset of the Cartesian product X × Y. The sets X and Y are called the domain (or the set of departure) andcodomain (or the set of destination), respectively, of the relation, and G is called its graph.The statement (x,y) ∈ G is read "x is R-related to y", and is denoted by xRy or R(x,y). The latter notation correspondsto viewing R as the characteristic function on X × Y for the set of pairs of G.The order of the elements in each pair of G is important: if a ≠ b, then aRb and bRa can be true or false, independentlyof each other. Resuming the above example, the prime 3 divides the integer 9, but 9 doesn't divide 3.A relation as defined by the triple (X, Y, G) is sometimes referred to as a correspondence instead.[1] In this case therelation from X to Y is the subset G of X × Y, and “from X to Y" must always be either specified or implied by thecontext when referring to the relation. In practice correspondence and relation tend to be used interchangeably.

33

https://en.wikipedia.org/wiki/Finitary_relation

https://en.wikipedia.org/wiki/Theory_of_relations

https://en.wikipedia.org/wiki/Relation#Mathematics



https://en.wikipedia.org/wiki/Ordered_pair


https://en.wikipedia.org/wiki/Cartesian_product

https://en.wikipedia.org/wiki/Divides

https://en.wikipedia.org/wiki/Prime_number





https://en.wikipedia.org/wiki/Arithmetic

https://en.wikipedia.org/wiki/Congruence_(geometry)

https://en.wikipedia.org/wiki/Geometry

https://en.wikipedia.org/wiki/Graph_theory

https://en.wikipedia.org/wiki/Orthogonal

https://en.wikipedia.org/wiki/Linear_algebra







https://en.wikipedia.org/wiki/Class_(mathematics)

https://en.wikipedia.org/wiki/Set_theory


https://en.wikipedia.org/wiki/Russell%2527s_paradox






https://en.wikipedia.org/wiki/Indicator_function

34 CHAPTER 11. HOMOGENEOUS RELATION

11.1.1 Is a relation more than its graph?

According to the definition above, two relations with identical graphs but different domains or different codomainsare considered different. For example, ifG = {(1, 2), (1, 3), (2, 7)} , then (Z,Z, G) , (R,N, G) , and (N,R, G) arethree distinct relations, where Z is the set of integers and R is the set of real numbers.Especially in set theory, binary relations are often defined as sets of ordered pairs, identifying binary relations withtheir graphs. The domain of a binary relation R is then defined as the set of all x such that there exists at least oney such that (x, y) ∈ R , the range of R is defined as the set of all y such that there exists at least one x such that(x, y) ∈ R , and the field of R is the union of its domain and its range.[2][3][4]

A special case of this difference in points of view applies to the notion of function. Many authors insist on distin-guishing between a function’s codomain and its range. Thus, a single “rule,” like mapping every real number x tox2, can lead to distinct functions f : R → R and f : R → R+ , depending on whether the images under thatrule are understood to be reals or, more restrictively, non-negative reals. But others view functions as simply sets ofordered pairs with unique first components. This difference in perspectives does raise some nontrivial issues. As anexample, the former camp considers surjectivity—or being onto—as a property of functions, while the latter sees itas a relationship that functions may bear to sets.Either approach is adequate for most uses, provided that one attends to the necessary changes in language, notation,and the definitions of concepts like restrictions, composition, inverse relation, and so on. The choice between the twodefinitions usually matters only in very formal contexts, like category theory.

11.1.2 Example

Example: Suppose there are four objects {ball, car, doll, gun} and four persons {John, Mary, Ian, Venus}. Supposethat John owns the ball, Mary owns the doll, and Venus owns the car. Nobody owns the gun and Ian owns nothing.Then the binary relation “is owned by” is given as

R = ({ball, car, doll, gun}, {John, Mary, Ian, Venus}, {(ball, John), (doll, Mary), (car, Venus)}).

Thus the first element of R is the set of objects, the second is the set of persons, and the last element is a set of orderedpairs of the form (object, owner).The pair (ball, John), denoted by ₐ RJₒ means that the ball is owned by John.Two different relations could have the same graph. For example: the relation

({ball, car, doll, gun}, {John, Mary, Venus}, {(ball, John), (doll, Mary), (car, Venus)})

is different from the previous one as everyone is an owner. But the graphs of the two relations are the same.Nevertheless, R is usually identified or even defined as G(R) and “an ordered pair (x, y) ∈ G(R)" is usually denoted as"(x, y) ∈ R".

11.2 Special types of binary relations

Some important types of binary relations R between two sets X and Y are listed below. To emphasize that X and Ycan be different sets, some authors call such binary relations heterogeneous.[5][6]

Uniqueness properties:

• injective (also called left-unique[7]): for all x and z in X and y in Y it holds that if xRy and zRy then x = z. Forexample, the green relation in the diagram is injective, but the red relation is not, as it relates e.g. both x = −5and z = +5 to y = 25.

• functional (also called univalent[8] or right-unique[7] or right-definite[9]): for all x in X, and y and z in Yit holds that if xRy and xRz then y = z; such a binary relation is called a partial function. Both relations inthe picture are functional. An example for a non-functional relation can be obtained by rotating the red graphclockwise by 90 degrees, i.e. by considering the relation x=y2 which relates e.g. x=25 to both y=−5 and z=+5.






https://en.wikipedia.org/wiki/Range_(mathematics)


https://en.wikipedia.org/wiki/Restriction_(mathematics)




https://en.wikipedia.org/wiki/Partial_function

11.2. SPECIAL TYPES OF BINARY RELATIONS 35

Example relations between real numbers. Red: y=x2. Green: y=2x+20.

• one-to-one (also written 1-to-1): injective and functional. The green relation is one-to-one, but the red is not.

Totality properties:

• left-total:[7] for all x in X there exists a y in Y such that xRy. For example R is left-total when it is a functionor a multivalued function. Note that this property, although sometimes also referred to as total, is differentfrom the definition of total in the next section. Both relations in the picture are left-total. The relation x=y2,obtained from the above rotation, is not left-total, as it doesn't relate, e.g., x = −14 to any real number y.

• surjective (also called right-total[7] or onto): for all y in Y there exists an x in X such that xRy. The greenrelation is surjective, but the red relation is not, as it doesn't relate any real number x to e.g. y = −14.

Uniqueness and totality properties:



• A function: a relation that is functional and left-total. Both the green and the red relation are functions.

• An injective function: a relation that is injective, functional, and left-total.

• A surjective function or surjection: a relation that is functional, left-total, and right-total.

• A bijection: a surjective one-to-one or surjective injective function is said to be bijective, also known asone-to-one correspondence.[10] The green relation is bijective, but the red is not.

11.2.1 Difunctional

Less commonly encountered is the notion of difunctional (or regular) relation, defined as a relation R such thatR=RR−1R.[11]

To understand this notion better, it helps to consider a relation as mapping every element x∈X to a set xR = { y∈Y| xRy }.[11] This set is sometimes called the successor neighborhood of x in R; one can define the predecessorneighborhood analogously.[12] Synonymous terms for these notions are afterset and respectively foreset.[5]

A difunctional relation can then be equivalently characterized as a relation R such that wherever x1R and x2R have anon-empty intersection, then these two sets coincide; formally x1R ∩ x2R ≠ ∅ implies x1R = x2R.[11]

As examples, any function or any functional (right-unique) relation is difunctional; the converse doesn't hold. If oneconsiders a relation R from set to itself (X = Y), then if R is both transitive and symmetric (i.e. a partial equivalencerelation), then it is also difunctional.[13] The converse of this latter statement also doesn't hold.A characterization of difunctional relations, which also explains their name, is to consider two functions f: A → Cand g: B → C and then define the following set which generalizes the kernel of a single function as joint kernel: ker(f,g) = { (a, b) ∈ A × B | f(a) = g(b) }. Every difunctional relation R ⊆ A × B arises as the joint kernel of two functionsf: A → C and g: B → C for some set C.[14]

In automata theory, the term rectangular relation has also been used to denote a difunctional relation. This ter-minology is justified by the fact that when represented as a boolean matrix, the columns and rows of a difunctionalrelation can be arranged in such a way as to present rectangular blocks of true on the (asymmetric) main diagonal.[15]

Other authors however use the term “rectangular” to denote any heterogeneous relation whatsoever.[6]

11.3 Relations over a set

If X = Y then we simply say that the binary relation is over X, or that it is an endorelation over X.[16] In computerscience, such a relation is also called a homogeneous (binary) relation.[16][17][6] Some types of endorelations arewidely studied in graph theory, where they are known as simple directed graphs permitting loops.The set of all binary relations Rel(X) on a set X is the set 2X × X which is a Boolean algebra augmented with theinvolution of mapping of a relation to its inverse relation. For the theoretical explanation see Relation algebra.Some important properties of a binary relation R over a set X are:

• reflexive: for all x in X it holds that xRx. For example, “greater than or equal to” (≥) is a reflexive relation but“greater than” (>) is not.

• irreflexive (or strict): for all x in X it holds that not xRx. For example, > is an irreflexive relation, but ≥ is not.

• coreflexive: for all x and y in X it holds that if xRy then x = y. An example of a coreflexive relation is therelation on integers in which each odd number is related to itself and there are no other relations. The equalityrelation is the only example of a both reflexive and coreflexive relation.

The previous 3 alternatives are far from being exhaustive; e.g. the red relation y=x2 from theabove picture is neither irreflexive, nor coreflexive, nor reflexive, since it contains the pair(0,0), and (2,4), but not (2,2), respectively.

• symmetric: for all x and y in X it holds that if xRy then yRx. “Is a blood relative of” is a symmetric relation,because x is a blood relative of y if and only if y is a blood relative of x.





https://en.wikipedia.org/wiki/Partial_equivalence_relation


https://en.wikipedia.org/wiki/Kernel_(set_theory)

https://en.wikipedia.org/wiki/Automata_theory


https://en.wikipedia.org/wiki/Directed_graph

https://en.wikipedia.org/wiki/Loop_(graph_theory)

https://en.wikipedia.org/wiki/Power_set

https://en.wikipedia.org/wiki/Boolean_algebra_(structure)

https://en.wikipedia.org/wiki/Involution_(mathematics)

https://en.wikipedia.org/wiki/Binary_relation#Operations_on_binary_relations



https://en.wikipedia.org/wiki/Coreflexive_relation

https://en.wikipedia.org/wiki/Binary_relation#Special_types_of_binary_relations


11.4. OPERATIONS ON BINARY RELATIONS 37

• antisymmetric: for all x and y in X, if xRy and yRx then x = y. For example, ≥ is anti-symmetric (so is >, butonly because the condition in the definition is always false).[18]

• asymmetric: for all x and y in X, if xRy then not yRx. A relation is asymmetric if and only if it is bothanti-symmetric and irreflexive.[19] For example, > is asymmetric, but ≥ is not.

• transitive: for all x, y and z in X it holds that if xRy and yRz then xRz. A transitive relation is irreflexive if andonly if it is asymmetric.[20] For example, “is ancestor of” is transitive, while “is parent of” is not.

• total: for all x and y in X it holds that xRy or yRx (or both). This definition for total is different from left totalin the previous section. For example, ≥ is a total relation.

• trichotomous: for all x and y in X exactly one of xRy, yRx or x = y holds. For example, > is a trichotomousrelation, while the relation “divides” on natural numbers is not.[21]

• Right Euclidean: for all x, y and z in X it holds that if xRy and xRz, then yRz.

• Left Euclidean: for all x, y and z in X it holds that if yRx and zRx, then yRz.

• Euclidean: An Euclidean relation is both left and right Euclidean. Equality is a Euclidean relation because ifx=y and x=z, then y=z.

• serial: for all x in X, there exists y in X such that xRy. "Is greater than" is a serial relation on the integers. Butit is not a serial relation on the positive integers, because there is no y in the positive integers (i.e. the naturalnumbers) such that 1>y.[22] However, "is less than" is a serial relation on the positive integers, the rationalnumbers and the real numbers. Every reflexive relation is serial: for a given x, choose y=x. A serial relation canbe equivalently characterized as every element having a non-empty successor neighborhood (see the previoussection for the definition of this notion). Similarly an inverse serial relation is a relation in which every elementhas non-empty predecessor neighborhood.[12]

• set-like (or local): for every x in X, the class of all y such that yRx is a set. (This makes sense only if relationson proper classes are allowed.) The usual ordering < on the class of ordinal numbers is set-like, while its inverse> is not.

A relation that is reflexive, symmetric, and transitive is called an equivalence relation. A relation that is symmetric,transitive, and serial is also reflexive. A relation that is only symmetric and transitive (without necessarily beingreflexive) is called a partial equivalence relation.A relation that is reflexive, antisymmetric, and transitive is called a partial order. A partial order that is total is calleda total order, simple order, linear order, or a chain.[23] A linear order where every nonempty subset has a least elementis called a well-order.

11.4 Operations on binary relations

If R, S are binary relations over X and Y, then each of the following is a binary relation over X and Y :

• Union: R ∪ S ⊆ X × Y, defined as R ∪ S = { (x, y) | (x, y) ∈ R or (x, y) ∈ S }. For example, ≥ is the union of >and =.

• Intersection: R ∩ S ⊆ X × Y, defined as R ∩ S = { (x, y) | (x, y) ∈ R and (x, y) ∈ S }.

If R is a binary relation over X and Y, and S is a binary relation over Y and Z, then the following is a binary relationover X and Z: (see main article composition of relations)

• Composition: S ∘ R, also denoted R ; S (or more ambiguously R ∘ S), defined as S ∘ R = { (x, z) | there existsy ∈ Y, such that (x, y) ∈ R and (y, z) ∈ S }. The order of R and S in the notation S ∘ R, used here agrees withthe standard notational order for composition of functions. For example, the composition “is mother of” ∘ “isparent of” yields “is maternal grandparent of”, while the composition “is parent of” ∘ “is mother of” yields “isgrandmother of”.






https://en.wikipedia.org/wiki/Trichotomy_(mathematics)




https://en.wikipedia.org/wiki/Binary_relation#difunctional


https://en.wikipedia.org/wiki/Class_(set_theory)

https://en.wikipedia.org/wiki/Ordinal_number

https://en.wikipedia.org/wiki/Equivalence_relation




https://en.wikipedia.org/wiki/Least_element



https://en.wikipedia.org/wiki/Composition_of_functions


A relation R on sets X and Y is said to be contained in a relation S on X and Y if R is a subset of S, that is, if x R yalways implies x S y. In this case, if R and S disagree, R is also said to be smaller than S. For example, > is containedin ≥.If R is a binary relation over X and Y, then the following is a binary relation over Y and X:

• Inverse or converse: R −1, defined as R −1 = { (y, x) | (x, y) ∈ R }. A binary relation over a set is equal to itsinverse if and only if it is symmetric. See also duality (order theory). For example, “is less than” (<) is theinverse of “is greater than” (>).

If R is a binary relation over X, then each of the following is a binary relation over X:

• Reflexive closure: R =, defined as R = = { (x, x) | x ∈ X } ∪ R or the smallest reflexive relation over X containingR. This can be proven to be equal to the intersection of all reflexive relations containing R.

• Reflexive reduction: R ≠, defined as R ≠ = R \ { (x, x) | x ∈ X } or the largest irreflexive relation over Xcontained in R.

• Transitive closure: R +, defined as the smallest transitive relation over X containing R. This can be seen to beequal to the intersection of all transitive relations containing R.

• Transitive reduction: R −, defined as a minimal relation having the same transitive closure as R.

• Reflexive transitive closure: R *, defined as R * = (R +) =, the smallest preorder containing R.

• Reflexive transitive symmetric closure: R ≡, defined as the smallest equivalence relation over X containingR.

11.4.1 Complement

If R is a binary relation over X and Y, then the following too:

• The complement S is defined as x S y if not x R y. For example, on real numbers, ≤ is the complement of >.

The complement of the inverse is the inverse of the complement.If X = Y, the complement has the following properties:

• If a relation is symmetric, the complement is too.

• The complement of a reflexive relation is irreflexive and vice versa.

• The complement of a strict weak order is a total preorder and vice versa.

The complement of the inverse has these same properties.

11.4.2 Restriction

The restriction of a binary relation on a set X to a subset S is the set of all pairs (x, y) in the relation for which x andy are in S.If a relation is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, total, trichotomous, a partialorder, total order, strict weak order, total preorder (weak order), or an equivalence relation, its restrictions are too.However, the transitive closure of a restriction is a subset of the restriction of the transitive closure, i.e., in generalnot equal. For example, restricting the relation "x is parent of y" to females yields the relation "x is mother ofthe woman y"; its transitive closure doesn't relate a woman with her paternal grandmother. On the other hand, thetransitive closure of “is parent of” is “is ancestor of"; its restriction to females does relate a woman with her paternalgrandmother.



https://en.wikipedia.org/wiki/Duality_(order_theory)


https://en.wikipedia.org/wiki/Irreflexive


https://en.wikipedia.org/wiki/Transitive_reduction

https://en.wikipedia.org/wiki/Preorder

https://en.wikipedia.org/wiki/Reflexive_transitive_symmetric_closure


https://en.wikipedia.org/wiki/Complement_(set_theory)

https://en.wikipedia.org/wiki/Strict_weak_order



https://en.wikipedia.org/wiki/Irreflexive_relation











https://en.wikipedia.org/wiki/Strict_weak_order#Total_preorders


11.5. SETS VERSUS CLASSES 39

Also, the various concepts of completeness (not to be confused with being “total”) do not carry over to restrictions.For example, on the set of real numbers a property of the relation "≤" is that every non-empty subset S of R with anupper bound in R has a least upper bound (also called supremum) in R. However, for a set of rational numbers thissupremum is not necessarily rational, so the same property does not hold on the restriction of the relation "≤" to theset of rational numbers.The left-restriction (right-restriction, respectively) of a binary relation between X and Y to a subset S of its domain(codomain) is the set of all pairs (x, y) in the relation for which x (y) is an element of S.

11.4.3 Algebras, categories, and rewriting systems

Various operations on binary endorelations can be treated as giving rise to an algebraic structure, known as relationalgebra. It should not be confused with relational algebra which deals in finitary relations (and in practice also finiteand many-sorted).For heterogenous binary relations, a category of relations arises.[6]

Despite their simplicity, binary relations are at the core of an abstract computation model known as an abstractrewriting system.

11.5 Sets versus classes

Certain mathematical “relations”, such as “equal to”, “member of”, and “subset of”, cannot be understood to be binaryrelations as defined above, because their domains and codomains cannot be taken to be sets in the usual systems ofaxiomatic set theory. For example, if we try to model the general concept of “equality” as a binary relation =, wemust take the domain and codomain to be the “class of all sets”, which is not a set in the usual set theory.In most mathematical contexts, references to the relations of equality, membership and subset are harmless becausethey can be understood implicitly to be restricted to some set in the context. The usual work-around to this problemis to select a “large enough” set A, that contains all the objects of interest, and work with the restriction =A instead of=. Similarly, the “subset of” relation ⊆ needs to be restricted to have domain and codomain P(A) (the power set ofa specific set A): the resulting set relation can be denoted ⊆A. Also, the “member of” relation needs to be restrictedto have domain A and codomain P(A) to obtain a binary relation ∈A that is a set. Bertrand Russell has shown thatassuming ∈ to be defined on all sets leads to a contradiction in naive set theory.Another solution to this problem is to use a set theory with proper classes, such as NBG or Morse–Kelley set theory,and allow the domain and codomain (and so the graph) to be proper classes: in such a theory, equality, membership,and subset are binary relations without special comment. (A minor modification needs to be made to the concept ofthe ordered triple (X, Y, G), as normally a proper class cannot be a member of an ordered tuple; or of course onecan identify the function with its graph in this context.)[24] With this definition one can for instance define a functionrelation between every set and its power set.

11.6 The number of binary relations

The number of distinct binary relations on an n-element set is 2n2 (sequence A002416 in OEIS):Notes:

• The number of irreflexive relations is the same as that of reflexive relations.

• The number of strict partial orders (irreflexive transitive relations) is the same as that of partial orders.

• The number of strict weak orders is the same as that of total preorders.

• The total orders are the partial orders that are also total preorders. The number of preorders that are neithera partial order nor a total preorder is, therefore, the number of preorders, minus the number of partial orders,minus the number of total preorders, plus the number of total orders: 0, 0, 0, 3, and 85, respectively.

• the number of equivalence relations is the number of partitions, which is the Bell number.

https://en.wikipedia.org/wiki/Completeness_(order_theory)


https://en.wikipedia.org/wiki/Empty_set

https://en.wikipedia.org/wiki/Upper_bound

https://en.wikipedia.org/wiki/Supremum

https://en.wikipedia.org/wiki/Algebraic_structure



https://en.wikipedia.org/wiki/Relational_algebra



https://en.wikipedia.org/wiki/Many-sorted


https://en.wikipedia.org/wiki/Abstract_rewriting_system




https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory

https://en.wikipedia.org/wiki/Morse%E2%80%93Kelley_set_theory

https://en.wikipedia.org/wiki/Proper_class

https://oeis.org/A002416

https://en.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequences

https://en.wikipedia.org/wiki/Partially_ordered_set#Strict_and_non-strict_partial_orders

https://en.wikipedia.org/wiki/Partition_of_a_set

https://en.wikipedia.org/wiki/Bell_number


The binary relations can be grouped into pairs (relation, complement), except that for n = 0 the relation is its owncomplement. The non-symmetric ones can be grouped into quadruples (relation, complement, inverse, inverse com-plement).

11.7 Examples of common binary relations

• order relations, including strict orders:

• greater than• greater than or equal to• less than• less than or equal to• divides (evenly)• is a subset of

• equivalence relations:

• equality• is parallel to (for affine spaces)• is in bijection with• isomorphy

• dependency relation, a finite, symmetric, reflexive relation.

• independency relation, a symmetric, irreflexive relation which is the complement of some dependency relation.

11.8 See also

• Confluence (term rewriting)

• Hasse diagram

• Incidence structure

• Logic of relatives

• Order theory

• Triadic relation

11.9 Notes[1] Encyclopedic dictionary of Mathematics. MIT. 2000. pp. 1330–1331. ISBN 0-262-59020-4.

[2] Suppes, Patrick (1972) [originally published by D. van Nostrand Company in 1960]. Axiomatic Set Theory. Dover. ISBN0-486-61630-4.

[3] Smullyan, Raymond M.; Fitting, Melvin (2010) [revised and corrected republication of the work originally published in1996 by Oxford University Press, New York]. Set Theory and the Continuum Problem. Dover. ISBN 978-0-486-47484-7.

[4] Levy, Azriel (2002) [republication of the work published by Springer-Verlag, Berlin, Heidelberg and New York in 1979].Basic Set Theory. Dover. ISBN 0-486-42079-5.

[5] Christodoulos A. Floudas; Panos M. Pardalos (2008). Encyclopedia of Optimization (2nd ed.). Springer Science & BusinessMedia. pp. 299–300. ISBN 978-0-387-74758-3.

https://en.wikipedia.org/wiki/Binary_relation#Complement

https://en.wikipedia.org/wiki/4-tuple



https://en.wikipedia.org/wiki/Strict_order

https://en.wikipedia.org/wiki/Greater_than

https://en.wikipedia.org/wiki/Less_than





https://en.wikipedia.org/wiki/Parallel_(geometry)

https://en.wikipedia.org/wiki/Affine_space



https://en.wikipedia.org/wiki/Dependency_relation

https://en.wikipedia.org/wiki/Independency_relation

https://en.wikipedia.org/wiki/Confluence_(term_rewriting)

https://en.wikipedia.org/wiki/Hasse_diagram

https://en.wikipedia.org/wiki/Incidence_structure

https://en.wikipedia.org/wiki/Logic_of_relatives


https://en.wikipedia.org/wiki/Triadic_relation

http://books.google.co.uk/books?id=azS2ktxrz3EC&pg=PA1331&hl=en&sa=X&ei=glo6T_PmC9Ow8QPvwYmFCw&ved=0CGIQ6AEwBg#v=onepage&f=false



https://en.wikipedia.org/wiki/Patrick_Suppes



https://en.wikipedia.org/wiki/Raymond_Smullyan



https://en.wikipedia.org/wiki/Azriel_Levy





11.10. REFERENCES 41

[6] Michael Winter (2007). Goguen Categories: A Categorical Approach to L-fuzzy Relations. Springer. pp. x–xi. ISBN978-1-4020-6164-6.

[7] Kilp, Knauer and Mikhalev: p. 3. The same four definitions appear in the following:

• Peter J. Pahl; Rudolf Damrath (2001). Mathematical Foundations of Computational Engineering: A Handbook.Springer Science & Business Media. p. 506. ISBN 978-3-540-67995-0.

• Eike Best (1996). Semantics of Sequential and Parallel Programs. Prentice Hall. pp. 19–21. ISBN 978-0-13-460643-9.

• Robert-Christoph Riemann (1999). Modelling of Concurrent Systems: Structural and Semantical Methods in the HighLevel Petri Net Calculus. Herbert Utz Verlag. pp. 21–22. ISBN 978-3-89675-629-9.

[8] Gunther Schmidt, 2010. Relational Mathematics. Cambridge University Press, ISBN 978-0-521-76268-7, Chapt. 5

[9] Mäs, Stephan (2007), “Reasoning on Spatial Semantic Integrity Constraints”, Spatial Information Theory: 8th InternationalConference, COSIT 2007, Melbourne, Australiia, September 19–23, 2007, Proceedings, Lecture Notes in Computer Science4736, Springer, pp. 285–302, doi:10.1007/978-3-540-74788-8_18

[10] Note that the use of “correspondence” here is narrower than as general synonym for binary relation.

[11] Chris Brink; Wolfram Kahl; Gunther Schmidt (1997). Relational Methods in Computer Science. Springer Science &Business Media. p. 200. ISBN 978-3-211-82971-4.

[12] Yao, Y. (2004). “Semantics of Fuzzy Sets in Rough Set Theory”. Transactions on Rough Sets II. Lecture Notes in ComputerScience 3135. p. 297. doi:10.1007/978-3-540-27778-1_15. ISBN 978-3-540-23990-1.

[13] William Craig (2006). Semigroups Underlying First-order Logic. American Mathematical Soc. p. 72. ISBN 978-0-8218-6588-0.

[14] Gumm, H. P.; Zarrad, M. (2014). “Coalgebraic Simulations and Congruences”. Coalgebraic Methods in Computer Science.Lecture Notes in Computer Science 8446. p. 118. doi:10.1007/978-3-662-44124-4_7. ISBN 978-3-662-44123-7.

[15] Julius Richard Büchi (1989). Finite Automata, Their Algebras and Grammars: Towards a Theory of Formal Expressions.Springer Science & Business Media. pp. 35–37. ISBN 978-1-4613-8853-1.

[16] M. E. Müller (2012). Relational Knowledge Discovery. Cambridge University Press. p. 22. ISBN 978-0-521-19021-3.

[17] Peter J. Pahl; Rudolf Damrath (2001). Mathematical Foundations of Computational Engineering: A Handbook. SpringerScience & Business Media. p. 496. ISBN 978-3-540-67995-0.

[18] Smith, Douglas; Eggen, Maurice; St. Andre, Richard (2006), ATransition to AdvancedMathematics (6th ed.), Brooks/Cole,p. 160, ISBN 0-534-39900-2


[20] Flaška, V.; Ježek, J.; Kepka, T.; Kortelainen, J. (2007). Transitive Closures of Binary Relations I (PDF). Prague: Schoolof Mathematics – Physics Charles University. p. 1. Lemma 1.1 (iv). This source refers to asymmetric relations as “strictlyantisymmetric”.

[21] Since neither 5 divides 3, nor 3 divides 5, nor 3=5.

[22] Yao, Y.Y.; Wong, S.K.M. (1995). “Generalization of rough sets using relationships between attribute values” (PDF).Proceedings of the 2nd Annual Joint Conference on Information Sciences: 30–33..

[23] Joseph G. Rosenstein, Linear orderings, Academic Press, 1982, ISBN 0-12-597680-1, p. 4

[24] Tarski, Alfred; Givant, Steven (1987). A formalization of set theory without variables. American Mathematical Society. p.3. ISBN 0-8218-1041-3.

11.10 References• M. Kilp, U. Knauer, A.V. Mikhalev, Monoids, Acts and Categories: with Applications to Wreath Products and

Graphs, De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, 2000, ISBN 3-11-015248-7.

• Gunther Schmidt, 2010. Relational Mathematics. Cambridge University Press, ISBN 978-0-521-76268-7.










https://en.wikipedia.org/wiki/Gunther_Schmidt



https://dx.doi.org/10.1007%252F978-3-540-74788-8_18




https://dx.doi.org/10.1007%252F978-3-540-27778-1_15







https://dx.doi.org/10.1007%252F978-3-662-44124-4_7



https://en.wikipedia.org/wiki/Julius_Richard_B%C3%BCchi











http://www2.cs.uregina.ca/~yyao/PAPERS/relation.pdf


https://en.wikipedia.org/wiki/Alfred_Tarski







11.11 External links• Hazewinkel, Michiel, ed. (2001), “Binary relation”, Encyclopedia of Mathematics, Springer, ISBN 978-1-

55608-010-4







Chapter 12

Cointerpretability

In mathematical logic, cointerpretability is a binary relation on formal theories: a formal theory T is cointerpretablein another such theory S, when the language of S can be translated into the language of T in such a way that S provesevery formula whose translation is a theorem of T. The “translation” here is required to preserve the logical structureof formulas.This concept, in a sense dual to interpretability, was introduced by Dzhaparidze (1993), who also proved that, fortheories of Peano arithmetic and any stronger theories with effective axiomatizations, cointerpretability is equivalentto Σ1 -conservativity.

12.1 See also• Cotolerance

• interpretability logic.

• Tolerance (in logic)

12.2 References• Dzhaparidze, Giorgie (1993), “A generalized notion of weak interpretability and the corresponding modal

logic”, Annals of Pure andApplied Logic 61 (1-2): 113–160, doi:10.1016/0168-0072(93)90201-N, MR 1218658.

• Japaridze, Giorgi; de Jongh, Dick (1998), “The logic of provability”, in Buss, Samuel R., Handbook of ProofTheory, Studies in Logic and the Foundations of Mathematics 137, Amsterdam: North-Holland, pp. 475–546,doi:10.1016/S0049-237X(98)80022-0, MR 1640331.

43



https://en.wikipedia.org/wiki/Theory_(mathematical_logic)

https://en.wikipedia.org/wiki/Theorem

https://en.wikipedia.org/wiki/Interpretability

https://en.wikipedia.org/wiki/Cointerpretability#CITEREFDzhaparidze1993

https://en.wikipedia.org/wiki/Peano_arithmetic

https://en.wikipedia.org/wiki/Axiomatization

https://en.wikipedia.org/wiki/Cotolerance

https://en.wikipedia.org/wiki/Interpretability_logic

https://en.wikipedia.org/wiki/Tolerance_(in_logic)

https://en.wikipedia.org/wiki/Giorgi_Japaridze


https://dx.doi.org/10.1016%252F0168-0072%252893%252990201-N



https://en.wikipedia.org/wiki/Giorgi_Japaridze

https://en.wikipedia.org/wiki/Dick_de_Jongh

https://en.wikipedia.org/wiki/Samuel_Buss


https://dx.doi.org/10.1016%252FS0049-237X%252898%252980022-0



Chapter 13

Commutative property

For other uses, see Commute (disambiguation).In mathematics, a binary operation is commutative if changing the order of the operands does not change the result.

=

==

This image illustrates that addition is commutative.

It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar asthe name of the property that says “3 + 4 = 4 + 3” or “2 × 5 = 5 × 2”, the property can also be used in more advancedsettings. The name is needed because there are operations, such as division and subtraction that do not have it (forexample, “3 − 5 ≠ 5 − 3”), such operations are not commutative, or noncommutative operations. The idea that simpleoperations, such as multiplication and addition of numbers, are commutative was for many years implicitly assumedand the property was not named until the 19th century when mathematics started to become formalized.

13.1 Common uses

The commutative property (or commutative law) is a property generally associated with binary operations and functions.If the commutative property holds for a pair of elements under a certain binary operation then the two elements aresaid to commute under that operation.

44

https://en.wikipedia.org/wiki/Commute_(disambiguation)


https://en.wikipedia.org/wiki/Binary_operation

https://en.wikipedia.org/wiki/Operand

https://en.wikipedia.org/wiki/Binary_operations

https://en.wikipedia.org/wiki/Mathematical_proof

https://en.wikipedia.org/wiki/Division_(mathematics)

https://en.wikipedia.org/wiki/Subtraction

https://en.wikipedia.org/wiki/Multiplication_(mathematics)

https://en.wikipedia.org/wiki/Addition


13.2. MATHEMATICAL DEFINITIONS 45

13.2 Mathematical definitions

Further information: Symmetric function

The term “commutative” is used in several related senses.[1][2]

1. A binary operation ∗ on a set S is called commutative if:x ∗ y = y ∗ x for all x, y ∈ S

An operation that does not satisfy the above property is called non-commutative.2. One says that x commutes with y under ∗ if:x ∗ y = y ∗ x

3. A binary function f : A×A→ B is called commutative if:f(x, y) = f(y, x) for all x, y ∈ A

13.3 Examples

13.3.1 Commutative operations in everyday life• Putting on socks resembles a commutative operation, since which sock is put on first is unimportant. Either

way, the result (having both socks on), is the same.• The commutativity of addition is observed when paying for an item with cash. Regardless of the order the bills

are handed over in, they always give the same total.

13.3.2 Commutative operations in mathematics

Two well-known examples of commutative binary operations:[1]

• The addition of real numbers is commutative, since

y + z = z + y for all y, z ∈ R

For example 4 + 5 = 5 + 4, since both expressions equal 9.

• The multiplication of real numbers is commutative, since

yz = zy for all y, z ∈ R

For example, 3 × 5 = 5 × 3, since both expressions equal 15.

• Some binary truth functions are also commutative, since the truth tables for the functions are the same whenone changes the order of the operands.

For example, the logical biconditional function p ↔ q is equivalent to q ↔ p. This function is also writtenas p IFF q, or as p ≡ q, or as Epq.The last form is an example of the most concise notation in the article on truth functions, which lists thesixteen possible binary truth functions of which eight are commutative: Vpq = Vqp; Apq (OR) = Aqp;Dpq (NAND) = Dqp; Epq (IFF) = Eqp; Jpq = Jqp; Kpq (AND) = Kqp; Xpq (NOR) = Xqp; Opq = Oqp.

• Further examples of commutative binary operations include addition and multiplication of complex numbers,addition and scalar multiplication of vectors, and intersection and union of sets.

https://en.wikipedia.org/wiki/Symmetric_function


https://en.wikipedia.org/wiki/Binary_function



https://en.wikipedia.org/wiki/Expression_(mathematics)

https://en.wikipedia.org/wiki/Multiplication


https://en.wikipedia.org/wiki/Truth_function

https://en.wikipedia.org/wiki/Truth_table

https://en.wikipedia.org/wiki/Logical_biconditional


https://en.wikipedia.org/wiki/Complex_number

https://en.wikipedia.org/wiki/Scalar_product





46 CHAPTER 13. COMMUTATIVE PROPERTY

b

b

aaa+

b

The addition of vectors is commutative, because a⃗+ b⃗ = b⃗+ a⃗ .

13.3.3 Noncommutative operations in everyday life

• Concatenation, the act of joining character strings together, is a noncommutative operation. For example

EA+ T = EAT ̸= TEA = T + EA

• Washing and drying clothes resembles a noncommutative operation; washing and then drying produces amarkedly different result to drying and then washing.

• Rotating a book 90° around a vertical axis then 90° around a horizontal axis produces a different orientationthan when the rotations are performed in the opposite order.

• The twists of the Rubik’s Cube are noncommutative. This can be studied using group theory.

• Also thought processes are noncommutative: A person asked a question (A) and then a question (B) may givedifferent answers to each question than a person asked first (B) and then (A), because asking a question maychange the person’s state of mind.

https://en.wikipedia.org/wiki/Concatenation

https://en.wikipedia.org/wiki/Rubik%2527s_Cube

https://en.wikipedia.org/wiki/Group_theory

13.4. HISTORY AND ETYMOLOGY 47

13.3.4 Noncommutative operations in mathematics

Some noncommutative binary operations:[3]

• Subtraction is noncommutative, since 0− 1 ̸= 1− 0

• Division is noncommutative, since 1/2 ̸= 2/1

• Some truth functions are noncommutative, since the truth tables for the functions are different when one changesthe order of the operands.

For example, the truth tables for f (A,B) = A Λ ¬B (A AND NOT B) and f (B,A) = B Λ ¬A are

• Matrix multiplication is noncommutative since

[0 20 1

]=

[1 10 1

]·[0 10 1

]̸=[0 10 1

]·[1 10 1

]=

[0 10 1

]• The vector product (or cross product) of two vectors in three dimensions is anti-commutative, i.e., b × a = −(a

× b).

13.4 History and etymology

The first known use of the term was in a French Journal published in 1814

Records of the implicit use of the commutative property go back to ancient times. The Egyptians used the commuta-tive property of multiplication to simplify computing products.[4][5] Euclid is known to have assumed the commutativeproperty of multiplication in his book Elements.[6] Formal uses of the commutative property arose in the late 18th andearly 19th centuries, when mathematicians began to work on a theory of functions. Today the commutative propertyis a well known and basic property used in most branches of mathematics.The first recorded use of the term commutative was in a memoir by François Servois in 1814,[7][8] which used theword commutatives when describing functions that have what is now called the commutative property. The word is acombination of the French word commuter meaning “to substitute or switch” and the suffix -ative meaning “tendingto” so the word literally means “tending to substitute or switch.” The term then appeared in English in PhilosophicalTransactions of the Royal Society in 1844.[7]

https://en.wikipedia.org/wiki/Subtraction

https://en.wikipedia.org/wiki/Division_(mathematics)



https://en.wikipedia.org/wiki/Matrix_(mathematics)

https://en.wikipedia.org/wiki/Cross_product

https://en.wikipedia.org/wiki/Anticommutativity

https://en.wikipedia.org/wiki/Egypt


https://en.wikipedia.org/wiki/Product_(mathematics)

https://en.wikipedia.org/wiki/Euclid

https://en.wikipedia.org/wiki/Euclid%2527s_Elements

https://en.wikipedia.org/wiki/Fran%C3%A7ois-Joseph_Servois

https://en.wikipedia.org/wiki/Philosophical_Transactions_of_the_Royal_Society

https://en.wikipedia.org/wiki/Philosophical_Transactions_of_the_Royal_Society


13.5 Propositional logic

13.5.1 Rule of replacement

In truth-functional propositional logic, commutation,[9][10] or commutativity[11] refer to two valid rules of replacement.The rules allow one to transpose propositional variables within logical expressions in logical proofs. The rules are:

(P ∨Q) ⇔ (Q ∨ P )

and

(P ∧Q) ⇔ (Q ∧ P )

where " ⇔ " is a metalogical symbol representing “can be replaced in a proof with.”

13.5.2 Truth functional connectives

Commutativity is a property of some logical connectives of truth functional propositional logic. The following logicalequivalences demonstrate that commutativity is a property of particular connectives. The following are truth-functionaltautologies.

Commutativity of conjunction (P ∧Q) ↔ (Q ∧ P )

Commutativity of disjunction (P ∨Q) ↔ (Q ∨ P )

Commutativity of implication (also called the law of permutation(P → (Q→ R)) ↔ (Q→ (P → R))

Commutativity of equivalence (also called the complete commutative law of equivalence)(P ↔ Q) ↔ (Q↔ P )

13.6 Set theory

In group and set theory, many algebraic structures are called commutative when certain operands satisfy the com-mutative property. In higher branches of mathematics, such as analysis and linear algebra the commutativity ofwell-known operations (such as addition and multiplication on real and complex numbers) is often used (or implicitlyassumed) in proofs.[12][13][14]

13.7 Mathematical structures and commutativity

• A commutative semigroup is a set endowed with a total, associative and commutative operation.

• If the operation additionally has an identity element, we have a commutative monoid

• An abelian group, or commutative group is a group whose group operation is commutative.[13]

• A commutative ring is a ring whose multiplication is commutative. (Addition in a ring is always commutative.)[15]

• In a field both addition and multiplication are commutative.[16]

https://en.wikipedia.org/wiki/Validity

https://en.wikipedia.org/wiki/Rule_of_replacement

https://en.wikipedia.org/wiki/Propositional_variable

https://en.wikipedia.org/wiki/Well-formed_formula

https://en.wikipedia.org/wiki/Formal_proof

https://en.wikipedia.org/wiki/Metalogic

https://en.wikipedia.org/wiki/Symbol_(formal)

https://en.wikipedia.org/wiki/Formal_proof

https://en.wikipedia.org/wiki/Logical_connective

https://en.wikipedia.org/wiki/Propositional_logic

https://en.wikipedia.org/wiki/Logical_equivalence


https://en.wikipedia.org/wiki/Tautology_(logic)



https://en.wikipedia.org/wiki/Mathematical_analysis




https://en.wikipedia.org/wiki/Commutative_semigroup

https://en.wikipedia.org/wiki/Associativity

https://en.wikipedia.org/wiki/Identity_element

https://en.wikipedia.org/wiki/Commutative_monoid



https://en.wikipedia.org/wiki/Commutative_ring




13.8. RELATED PROPERTIES 49

13.8 Related properties

13.8.1 Associativity

Main article: Associative property

The associative property is closely related to the commutative property. The associative property of an expressioncontaining two or more occurrences of the same operator states that the order operations are performed in does notaffect the final result, as long as the order of terms doesn't change. In contrast, the commutative property states thatthe order of the terms does not affect the final result.Most commutative operations encountered in practice are also associative. However, commutativity does not implyassociativity. A counterexample is the function

f(x, y) =x+ y

2,

which is clearly commutative (interchanging x and y does not affect the result), but it is not associative (since, forexample, f(−4, f(0,+4)) = −1 but f(f(−4, 0),+4) = +1 ). More such examples may be found in Commutativenon-associative magmas.

13.8.2 Symmetry

-10-8-6-4-2 0 2 4 6 8 10

Graph showing the symmetry of the addition function

Main article: Symmetry in mathematics

https://en.wikipedia.org/wiki/Associative_property

https://en.wikipedia.org/wiki/Commutative_non-associative_magmas

https://en.wikipedia.org/wiki/Commutative_non-associative_magmas



Some forms of symmetry can be directly linked to commutativity. When a commutative operator is written as abinary function then the resulting function is symmetric across the line y = x. As an example, if we let a function frepresent addition (a commutative operation) so that f(x,y) = x + y then f is a symmetric function, which can be seenin the image on the right.For relations, a symmetric relation is analogous to a commutative operation, in that if a relation R is symmetric, thenaRb⇔ bRa .

13.9 Non-commuting operators in quantum mechanics

Main article: Canonical commutation relation

In quantum mechanics as formulated by Schrödinger, physical variables are represented by linear operators such as x(meaning multiply by x), and d

dx . These two operators do not commute as may be seen by considering the effect oftheir compositions x d

dx and ddxx (also called products of operators) on a one-dimensional wave function ψ(x) :

xd

dxψ = xψ′ ̸= d

dxxψ = ψ + xψ′

According to the uncertainty principle of Heisenberg, if the two operators representing a pair of variables do notcommute, then that pair of variables are mutually complementary, which means they cannot be simultaneously mea-sured or known precisely. For example, the position and the linear momentum in the x-direction of a particle arerepresented respectively by the operators x and −iℏ ∂

∂x (where ℏ is the reduced Planck constant). This is the sameexample except for the constant −iℏ , so again the operators do not commute and the physical meaning is that theposition and linear momentum in a given direction are complementary.

13.10 See also

• Anticommutativity

• Associative Property

• Binary operation

• Centralizer or Commutant

• Commutative diagram

• Commutative (neurophysiology)

• Commutator

• Distributivity

• Parallelogram law

• Particle statistics (for commutativity in physics)

• Quasi-commutative property

• Trace monoid

• Truth function

• Truth table


https://en.wikipedia.org/wiki/Canonical_commutation_relation

https://en.wikipedia.org/wiki/Introduction_to_quantum_mechanics

https://en.wikipedia.org/wiki/Erwin_Schr%C3%B6dinger

https://en.wikipedia.org/wiki/Linear_operators

https://en.wikipedia.org/wiki/Function_composition

https://en.wikipedia.org/wiki/Wave_function

https://en.wikipedia.org/wiki/Uncertainty_principle

https://en.wikipedia.org/wiki/Werner_Heisenberg

https://en.wikipedia.org/wiki/Complementarity_(physics)

https://en.wikipedia.org/wiki/Momentum

https://en.wikipedia.org/wiki/Planck_constant

https://en.wikipedia.org/wiki/Anticommutativity

https://en.wikipedia.org/wiki/Associative_Property


https://en.wikipedia.org/wiki/Centralizer

https://en.wikipedia.org/wiki/Commutant

https://en.wikipedia.org/wiki/Commutative_diagram

https://en.wikipedia.org/wiki/Commutative_(neurophysiology)

https://en.wikipedia.org/wiki/Commutator

https://en.wikipedia.org/wiki/Distributivity

https://en.wikipedia.org/wiki/Parallelogram_law

https://en.wikipedia.org/wiki/Particle_statistics

https://en.wikipedia.org/wiki/Physics

https://en.wikipedia.org/wiki/Quasi-commutative_property

https://en.wikipedia.org/wiki/Trace_monoid



13.11. NOTES 51

13.11 Notes

[1] Krowne, p.1

[2] Weisstein, Commute, p.1

[3] Yark, p.1

[4] Lumpkin, p.11

[5] Gay and Shute, p.?

[6] O'Conner and Robertson, Real Numbers

[7] Cabillón and Miller, Commutative and Distributive

[8] O'Conner and Robertson, Servois

[9] Moore and Parker

[10] Copi, Irving M.; Cohen, Carl (2005). Introduction to Logic. Prentice Hall.

[11] Hurley, Patrick (1991). A Concise Introduction to Logic 4th edition. Wadsworth Publishing.

[12] Axler, p.2

[13] Gallian, p.34

[14] p. 26,87

[15] Gallian p.236

[16] Gallian p.250

13.12 References

13.12.1 Books

• Axler, Sheldon (1997). Linear Algebra Done Right, 2e. Springer. ISBN 0-387-98258-2.

Abstract algebra theory. Covers commutativity in that context. Uses property throughout book.

• Copi, Irving M.; Cohen, Carl (2005). Introduction to Logic. Prentice Hall.

• Gallian, Joseph (2006). Contemporary Abstract Algebra, 6e. Boston, Mass.: Houghton Mifflin. ISBN 0-618-51471-6.

Linear algebra theory. Explains commutativity in chapter 1, uses it throughout.

• Goodman, Frederick (2003). Algebra: Abstract and Concrete, Stressing Symmetry, 2e. Prentice Hall. ISBN0-13-067342-0.

Abstract algebra theory. Uses commutativity property throughout book.

• Hurley, Patrick (1991). A Concise Introduction to Logic 4th edition. Wadsworth Publishing.









13.12.2 Articles

• http://www.ethnomath.org/resources/lumpkin1997.pdf Lumpkin, B. (1997). The Mathematical Legacy OfAncient Egypt - A Response To Robert Palter. Unpublished manuscript.

Article describing the mathematical ability of ancient civilizations.

• Robins, R. Gay, and Charles C. D. Shute. 1987. The Rhind Mathematical Papyrus: An Ancient Egyptian Text.London: British Museum Publications Limited. ISBN 0-7141-0944-4

Translation and interpretation of the Rhind Mathematical Papyrus.

13.12.3 Online resources

• Hazewinkel, Michiel, ed. (2001), “Commutativity”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

• Krowne, Aaron, Commutative at PlanetMath.org., Accessed 8 August 2007.

Definition of commutativity and examples of commutative operations

• Weisstein, Eric W., “Commute”, MathWorld., Accessed 8 August 2007.

Explanation of the term commute

• Yark. Examples of non-commutative operations at PlanetMath.org., Accessed 8 August 2007

Examples proving some noncommutative operations

• O'Conner, J J and Robertson, E F. MacTutor history of real numbers, Accessed 8 August 2007

Article giving the history of the real numbers

• Cabillón, Julio and Miller, Jeff. Earliest Known Uses Of Mathematical Terms, Accessed 22 November 2008

Page covering the earliest uses of mathematical terms

• O'Conner, J J and Robertson, E F. MacTutor biography of François Servois, Accessed 8 August 2007

Biography of Francois Servois, who first used the term

http://www.ethnomath.org/resources/lumpkin1997.pdf


https://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrus

http://www.encyclopediaofmath.org/index.php?title=p/c023420






http://planetmath.org/Commutative

https://en.wikipedia.org/wiki/PlanetMath


http://mathworld.wolfram.com/Commute.html


http://planetmath.org/?op=getuser&id=2760

http://planetmath.org/ExampleOfCommutative


http://www-history.mcs.st-andrews.ac.uk/HistTopics/Real_numbers_1.html

http://jeff560.tripod.com/c.html

http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Servois.html

Chapter 14

Comparability

See also: Comparison (mathematics)

In mathematics, any two elements x and y of a set P that is partially ordered by a binary relation ≤ are comparablewhen either x ≤ y or y ≤ x. If it is not the case that x and y are comparable, then they are called incomparable.A totally ordered set is exactly a partially ordered set in which every pair of elements is comparable.It follows immediately from the definitions of comparability and incomparability that both relations are symmetric,that is x is comparable to y if and only if y is comparable to x, and likewise for incomparability.

14.1 Notation

Comparability is denoted by the symbol ⊥, and incomparability by the symbol ||.[1] Thus, for any pair of elements xand y of a partially ordered set, exactly one of

• x ⊥ y and

• x || y

is true.

14.2 Comparability graphs

The comparability graph of a partially ordered set P has as vertices the elements of P and has as edges preciselythose pairs {x, y} of elements for which x ⊥ y.[2]

14.3 Classification

When classifying mathematical objects (e.g., topological spaces), two criteria are said to be comparable when theobjects that obey one criterion constitute a subset of the objects that obey the other, which is to say when they arecomparable under the partial order ⊂. For example, the T1 and T2 criteria are comparable, while the T1 and sobrietycriteria are not.

14.4 See also

• Strict weak ordering, a partial ordering in which incomparability is a transitive relation

53

https://en.wikipedia.org/wiki/Comparison_(mathematics)





https://en.wikipedia.org/wiki/Comparability_graph


https://en.wikipedia.org/wiki/Topological_space

https://en.wikipedia.org/wiki/T1_space

https://en.wikipedia.org/wiki/Hausdorff_space

https://en.wikipedia.org/wiki/Sober_space

https://en.wikipedia.org/wiki/Strict_weak_ordering

54 CHAPTER 14. COMPARABILITY

14.5 References

“PlanetMath: partial order”. Retrieved 6 April 2010.

[1] Trotter, William T. (1992), Combinatorics and Partially Ordered Sets:Dimension Theory, Johns Hopkins Univ. Press, p. 3

[2] Gilmore, P. C.; Hoffman, A. J. (1964), “A characterization of comparability graphs and of interval graphs”, CanadianJournal of Mathematics 16: 539–548, doi:10.4153/CJM-1964-055-5.

http://planetmath.org/encyclopedia/PartialOrder.html

https://en.wikipedia.org/wiki/Alan_Hoffman_(mathematician)

http://www.cms.math.ca/cjm/v16/p539


https://dx.doi.org/10.4153%252FCJM-1964-055-5

Chapter 15

Composition of relations

In mathematics, the composition of binary relations is a concept of forming a new relation S ∘ R from two givenrelations R and S, having as its most well-known special case the composition of functions.

15.1 Definition

If R ⊆ X × Y and S ⊆ Y × Z are two binary relations, then their composition S ◦R is the relation

S ◦R = {(x, z) ∈ X × Z | ∃y ∈ Y : (x, y) ∈ R ∧ (y, z) ∈ S}.

In other words, S ◦ R ⊆ X × Z is defined by the rule that says (x, z) ∈ S ◦ R if and only if there is an elementy ∈ Y such that xR y S z (i.e. (x, y) ∈ R and (y, z) ∈ S ).In particular fields, authors might denote by R ∘ S what is defined here to be S ∘ R. The convention chosen here issuch that function composition (with the usual notation) is obtained as a special case, when R and S are functionalrelations. Some authors[1] prefer to write ◦l and ◦r explicitly when necessary, depending whether the left or the rightrelation is the first one applied.A further variation encountered in computer science is the Z notation: ◦ is used to denote the traditional (right)composition, but ; (a fat open semicolon with Unicode code point U+2A3E) denotes left composition.[2][3] This useof semicolon coincides with the notation for function composition used (mostly by computer scientists) in Categorytheory,[4] as well as the notation for dynamic conjunction within linguistic dynamic semantics.[5] The semicolonnotation (with this semantic) was introduced by Ernst Schröder in 1895.[6]

The binary relations R ⊆ X × Y are sometimes regarded as the morphisms R : X → Y in a category Rel whichhas the sets as objects. In Rel, composition of morphisms is exactly composition of relations as defined above. Thecategory Set of sets is a subcategory of Rel that has the same objects but fewer morphisms. A generalization of thisis found in the theory of allegories.

15.2 Properties

Composition of relations is associative.The inverse relation of S ∘ R is (S ∘ R)−1 = R−1 ∘ S−1. This property makes the set of all binary relations on a set asemigroup with involution.The compose of (partial) functions (i.e. functional relations) is again a (partial) function.If R and S are injective, then S ∘ R is injective, which conversely implies only the injectivity of R.If R and S are surjective, then S ∘ R is surjective, which conversely implies only the surjectivity of S.The set of binary relations on a set X (i.e. relations from X to X) together with (left or right) relation compositionforms a monoid with zero, where the identity map on X is the neutral element, and the empty set is the zero element.

55





https://en.wikipedia.org/wiki/Functional_relation


https://en.wikipedia.org/wiki/Z_notation

https://en.wikipedia.org/wiki/Function_composition#Alternative_notations



https://en.wikipedia.org/wiki/Ernst_Schr%C3%B6der




https://en.wikipedia.org/wiki/Allegory_(category_theory)

https://en.wikipedia.org/wiki/Associative


https://en.wikipedia.org/wiki/Semigroup_with_involution


https://en.wikipedia.org/wiki/Injective

https://en.wikipedia.org/wiki/Surjective

https://en.wikipedia.org/wiki/Monoid

https://en.wikipedia.org/wiki/Neutral_element

https://en.wikipedia.org/wiki/Absorbing_element

56 CHAPTER 15. COMPOSITION OF RELATIONS

15.3 Join: another form of composition

Main article: Join (relational algebra)

Other forms of composition of relations, which apply to general n-place relations instead of binary relations, arefound in the join operation of relational algebra. The usual composition of two binary relations as defined here canbe obtained by taking their join, leading to a ternary relation, followed by a projection that removes the middlecomponent.

15.4 See also• Binary relation

• Relation algebra

• Demonic composition

• Function composition

• Join (SQL)

• Logical matrix

15.5 Notes[1] Kilp, Knauer & Mikhalev, p. 7

[2] ISO/IEC 13568:2002(E), p. 23

[3] http://www.fileformat.info/info/unicode/char/2a3e/index.htm

[4] http://www.math.mcgill.ca/triples/Barr-Wells-ctcs.pdf, p. 6

[5] http://plato.stanford.edu/entries/dynamic-semantics/#EncDynTypLog

[6] Paul Taylor (1999). Practical Foundations of Mathematics. Cambridge University Press. p. 24. ISBN 978-0-521-63107-5.A free HTML version of the book is available at http://www.cs.man.ac.uk/~{}pt/Practical_Foundations/

15.6 References• M. Kilp, U. Knauer, A.V. Mikhalev, Monoids, Acts and Categories with Applications to Wreath Products and


https://en.wikipedia.org/wiki/Join_(relational_algebra)

https://en.wikipedia.org/wiki/Join_(relational_algebra)




https://en.wikipedia.org/wiki/Demonic_composition


https://en.wikipedia.org/wiki/Join_(SQL)

https://en.wikipedia.org/wiki/Logical_matrix

http://www.fileformat.info/info/unicode/char/2a3e/index.htm

http://www.math.mcgill.ca/triples/Barr-Wells-ctcs.pdf

http://plato.stanford.edu/entries/dynamic-semantics/#EncDynTypLog

http://books.google.com/books?id=iSCqyNgzamcC&pg=PA24



http://www.cs.man.ac.uk/~pt/Practical_Foundations/


Chapter 16

Congruence relation

For the term as used in elementary geometry, see congruence (geometry).

In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic struc-ture (such as a group, ring, or vector space) that is compatible with the structure. Every congruence relation has acorresponding quotient structure, whose elements are the equivalence classes (or congruence classes) for the relation.

16.1 Basic example

The prototypical example of a congruence relation is congruence modulo n on the set of integers. For a given positiveinteger n , two integers a and b are called congruent modulo n , written

a ≡ b (mod n)

if a− b is divisible by n (or equivalently if a and b have the same remainder when divided by n ).for example, 37 and 57 are congruent modulo 10 ,

37 ≡ 57 (mod 10)

since 37 − 57 = −20 is a multiple of 10, or equivalently since both 37 and 57 have a remainder of 7 when dividedby 10 .Congruence modulo n (for a fixed n ) is compatible with both addition and multiplication on the integers. That is,if

a1 ≡ a2 (mod n) and b1 ≡ b2 (mod n)

then

a1 + b1 ≡ a2 + b2 (mod n) and a1b1 ≡ a2b2 (mod n)

The corresponding addition and multiplication of equivalence classes is known as modular arithmetic. From the pointof view of abstract algebra, congruence modulo n is a congruence relation on the ring of integers, and arithmeticmodulo n occurs on the corresponding quotient ring.

16.2 Definition

The definition of a congruence depends on the type of algebraic structure under consideration. Particular definitions ofcongruence can be made for groups, rings, vector spaces, modules, semigroups, lattices, and so forth. The common

57


https://en.wikipedia.org/wiki/Abstract_algebra







https://en.wikipedia.org/wiki/Quotient

https://en.wikipedia.org/wiki/Equivalence_class


https://en.wikipedia.org/wiki/Positive_integer


https://en.wikipedia.org/wiki/Divisible

https://en.wikipedia.org/wiki/Remainder



https://en.wikipedia.org/wiki/Modular_arithmetic


https://en.wikipedia.org/wiki/Quotient_ring






https://en.wikipedia.org/wiki/Semigroup

https://en.wikipedia.org/wiki/Lattice_(order)

58 CHAPTER 16. CONGRUENCE RELATION

theme is that a congruence is an equivalence relation on an algebraic object that is compatible with the algebraicstructure, in the sense that the operations are well-defined on the equivalence classes.For example, a group is an algebraic object consisting of a set together with a single binary operation, satisfyingcertain axioms. If G is a group with operation ∗, a congruence relation on G is an equivalence relation ≡ on theelements of G satisfying

g1 ≡ g2 and h1 ≡ h2 ⇒ g1 ∗ h1 ≡ g2 ∗ h2

for all g1, g2, h1, h2 ∈ G. For a congruence on a group, the equivalence class containing the identity element is alwaysa normal subgroup, and the other equivalence classes are the cosets of this subgroup. Together, these equivalenceclasses are the elements of a quotient group.When an algebraic structure includes more than one operation, congruence relations are required to be compatiblewith each operation. For example, a ring possesses both addition and multiplication, and a congruence relation on aring must satisfy

r1 + s1 ≡ r2 + s2 and r1s1 ≡ r2s2

whenever r1 ≡ r2 and s1 ≡ s2. For a congruence on a ring, the equivalence class containing 0 is always a two-sidedideal, and the two operations on the set of equivalence classes define the corresponding quotient ring.The general notion of a congruence relation can be given a formal definition in the context of universal algebra, afield which studies ideas common to all algebraic structures. In this setting, a congruence relation is an equivalencerelation ≡ on an algebraic structure that satisfies

μ(a1, a2, ..., an) ≡ μ(a1′, a2′, ..., an′)

for every n-ary operation μ, and all elements a1,...,an,a1′,...,an′ satisfying ai ≡ ai′ for each i.

16.3 Relation with homomorphisms

If ƒ: A → B is a homomorphism between two algebraic structures (such as homomorphism of groups, or a linear mapbetween vector spaces), then the relation R defined by

a1 R a2 if and only if ƒ(a1) = ƒ(a2)

is a congruence relation. By the first isomorphism theorem, the image of A under ƒ is a substructure of B isomorphicto the quotient of A by this congruence.

16.4 Congruences of groups, and normal subgroups and ideals

In the particular case of groups, congruence relations can be described in elementary terms as follows: If G is a group(with identity element e and operation *) and ~ is a binary relation on G, then ~ is a congruence whenever:

1. Given any element a of G, a ~ a (reflexivity);

2. Given any elements a and b of G, if a ~ b, then b ~ a (symmetry);

3. Given any elements a, b, and c of G, if a ~ b and b ~ c, then a ~ c (transitivity);

4. Given any elements a, a' , b, and b' of G, if a ~ a' and b ~ b' , then a * b ~ a' * b' ;

5. Given any elements a and a' of G, if a ~ a' , then a−1 ~ a' −1 (this can actually be proven from the other four,so is strictly redundant).


https://en.wikipedia.org/wiki/Operation_(mathematics)

https://en.wikipedia.org/wiki/Well-defined

https://en.wikipedia.org/wiki/Equivalence_classes




https://en.wikipedia.org/wiki/Normal_subgroup

https://en.wikipedia.org/wiki/Coset

https://en.wikipedia.org/wiki/Quotient_group

https://en.wikipedia.org/wiki/Ideal_(ring_theory)


https://en.wikipedia.org/wiki/Universal_algebra

https://en.wikipedia.org/wiki/Algebraic_structures


https://en.wikipedia.org/wiki/Group_homomorphism

https://en.wikipedia.org/wiki/Linear_map


https://en.wikipedia.org/wiki/First_isomorphism_theorem






https://en.wikipedia.org/wiki/Given_any


https://en.wikipedia.org/wiki/Material_conditional


https://en.wikipedia.org/wiki/Logical_conjunction


16.5. UNIVERSAL ALGEBRA 59

Conditions 1, 2, and 3 say that ~ is an equivalence relation.A congruence ~ is determined entirely by the set {a ∈ G : a ~ e} of those elements of G that are congruent to theidentity element, and this set is a normal subgroup. Specifically, a ~ b if and only if b−1 * a ~ e. So instead of talkingabout congruences on groups, people usually speak in terms of normal subgroups of them; in fact, every congruencecorresponds uniquely to some normal subgroup of G.

16.4.1 Ideals of rings and the general case

A similar trick allows one to speak of kernels in ring theory as ideals instead of congruence relations, and in moduletheory as submodules instead of congruence relations.The most general situation where this trick is possible is with Omega-groups (in the general sense allowing operatorswith multiple arity). But this cannot be done with, for example, monoids, so the study of congruence relations playsa more central role in monoid theory.

16.5 Universal algebra

The idea is generalized in universal algebra: A congruence relation on an algebra A is a subset of the direct productA × A that is both an equivalence relation on A and a subalgebra of A × A.The kernel of a homomorphism is always a congruence. Indeed, every congruence arises as a kernel. For a givencongruence ~ on A, the set A/~ of equivalence classes can be given the structure of an algebra in a natural fashion,the quotient algebra. The function that maps every element of A to its equivalence class is a homomorphism, and thekernel of this homomorphism is ~.The lattice Con(A) of all congruence relations on an algebra A is algebraic.

16.6 See also• Table of congruences

• Linear congruence theorem

• Congruence lattice problem

16.7 References• Horn and Johnson, Matrix Analysis, Cambridge University Press, 1985. ISBN 0-521-38632-2. (Section 4.5

discusses congruency of matrices.)




https://en.wikipedia.org/wiki/Ideal_(ring_theory)



https://en.wikipedia.org/wiki/Submodule

https://en.wikipedia.org/wiki/Omega-group




https://en.wikipedia.org/wiki/Direct_product


https://en.wikipedia.org/wiki/Subalgebra

https://en.wikipedia.org/wiki/Kernel_(universal_algebra)



https://en.wikipedia.org/wiki/Quotient_algebra


https://en.wikipedia.org/wiki/Algebraic_lattice

https://en.wikipedia.org/wiki/Table_of_congruences

https://en.wikipedia.org/wiki/Linear_congruence_theorem

https://en.wikipedia.org/wiki/Congruence_lattice_problem


Chapter 17

Contour set

In mathematics, contour sets generalize and formalize the everyday notions of

• everything superior to something

• everything superior or equivalent to something

• everything inferior to something

• everything inferior or equivalent to something.

17.1 Formal definitions

Given a relation on pairs of elements of set X

≽ ⊆ X2

and an element x of X

x ∈ X

The upper contour set of x is the set of all y that are related to x :

{y ϶ y ≽ x}

The lower contour set of x is the set of all y such that x is related to them:

{y ϶ x ≽ y}

The strict upper contour set of x is the set of all y that are related to x without x being in this way related to anyof them:

{y ϶ (y ≽ x) ∧ ¬(x ≽ y)}

The strict lower contour set of x is the set of all y such that x is related to them without any of them being in thisway related to x :

{y ϶ (x ≽ y) ∧ ¬(y ≽ x)}

60


https://en.wikipedia.org/wiki/Generalization

https://en.wikipedia.org/wiki/Formal_system




17.2. EXAMPLES 61

The formal expressions of the last two may be simplified if we have defined

≻ = {(a, b) ϶ (a ≽ b) ∧ ¬(b ≽ a)}

so that a is related to b but b is not related to a , in which case the strict upper contour set of x is

{y ϶ y ≻ x}

and the strict lower contour set of x is

{y ϶ x ≻ y}

17.1.1 Contour sets of a function

In the case of a function f() considered in terms of relation ▷ , reference to the contour sets of the function is implicitlyto the contour sets of the implied relation

(a ≽ b) ⇐ [f(a) ▷ f(b)]

17.2 Examples

17.2.1 Arithmetic

Consider a real number x , and the relation ≥ . Then

• the upper contour set of x would be the set of numbers that were greater than or equal to x ,

• the strict upper contour set of x would be the set of numbers that were greater than x ,

• the lower contour set of x would be the set of numbers that were less than or equal to x , and

• the strict lower contour set of x would be the set of numbers that were less than x .

Consider, more generally, the relation

(a ≽ b) ⇐ [f(a) ≥ f(b)]

Then

• the upper contour set of x would be the set of all y such that f(y) ≥ f(x) ,

• the strict upper contour set of x would be the set of all y such that f(y) > f(x) ,

• the lower contour set of x would be the set of all y such that f(x) ≥ f(y) , and

• the strict lower contour set of x would be the set of all y such that f(x) > f(y) .

It would be technically possible to define contour sets in terms of the relation

(a ≽ b) ⇐ [f(a) ≤ f(b)]



62 CHAPTER 17. CONTOUR SET

though such definitions would tend to confound ready understanding.In the case of a real-valued function f() (whose arguments might or might not be themselves real numbers), referenceto the contour sets of the function is implicitly to the contour sets of the relation

(a ≽ b) ⇐ [f(a) ≥ f(b)]

Note that the arguments to f() might be vectors, and that the notation used might instead be

[(a1, a2, . . .) ≽ (b1, b2, . . .)] ⇐ [f(a1, a2, . . .) ≥ f(b1, b2, . . .)]

17.2.2 Economic

In economics, the set X could be interpreted as a set of goods and services or of possible outcomes, the relation ≻as strict preference, and the relationship ≽ as weak preference. Then

• the upper contour set, or better set,[1] of x would be the set of all goods, services, or outcomes that were atleast as desired as x ,

• the strict upper contour set of x would be the set of all goods, services, or outcomes that were more desiredthan x ,

• the lower contour set, or worse set,[1] of x would be the set of all goods, services, or outcomes that were nomore desired than x , and

• the strict lower contour set of x would be the set of all goods, services, or outcomes that were less desired thanx .

Such preferences might be captured by a utility function u() , in which case

• the upper contour set of x would be the set of all y such that u(y) ≥ u(x) ,

• the strict upper contour set of x would be the set of all y such that u(y) > u(x) ,

• the lower contour set of x would be the set of all y such that u(x) ≥ u(y) , and

• the strict lower contour set of x would be the set of all y such that u(x) > u(y) .

17.3 Complementarity

On the assumption that ≽ is a total ordering ofX , the complement of the upper contour set is the strict lower contourset.

X2\ {y ϶ y ≽ x} = {y ϶ x ≻ y}

X2\ {y ϶ x ≻ y} = {y ϶ y ≽ x}

and the complement of the strict upper contour set is the lower contour set.

X2\ {y ϶ y ≻ x} = {y ϶ x ≽ y}

X2\ {y ϶ x ≽ y} = {y ϶ y ≻ x}



https://en.wikipedia.org/wiki/Economics

https://en.wikipedia.org/wiki/Good_(economics_and_accounting)

https://en.wikipedia.org/wiki/Outcome_(game_theory)

https://en.wikipedia.org/wiki/Preference

https://en.wikipedia.org/wiki/Utility



17.4. SEE ALSO 63

17.4 See also• Epigraph

• Hypograph

17.5 References[1] Robert P. Gilles (1996). Economic Exchange and Social Organization: The Edgeworthian Foundations of General Equilib-

rium Theory. Springer. p. 35.

17.6 Bibliography• Andreu Mas-Colell, Michael D. Whinston, and Jerry R. Green, Microeconomic Theory (LCC HB172.M6247

1995), p43. ISBN 0-19-507340-1 (cloth) ISBN 0-19-510268-1 (paper)

https://en.wikipedia.org/wiki/Epigraph_(mathematics)

https://en.wikipedia.org/wiki/Hypograph_(mathematics)

http://books.google.com/books?id=ZyahaTvMB3cC&lpg=PA35&ots=4CelGh9izH&dq=%2522better%2520set%2522%2520economics&pg=PA35#v=onepage&q=%2522better%2520set%2522%2520

http://books.google.com/books?id=ZyahaTvMB3cC&lpg=PA35&ots=4CelGh9izH&dq=%2522better%2520set%2522%2520economics&pg=PA35#v=onepage&q=%2522better%2520set%2522%2520

https://en.wikipedia.org/wiki/Andreu_Mas-Colell

https://en.wikipedia.org/wiki/Library_of_Congress_Classification

http://catalog.loc.gov/cgi-bin/Pwebrecon.cgi?Search_Arg=HB172.M6247+1995&Search_Code=CALL_&CNT=5

http://catalog.loc.gov/cgi-bin/Pwebrecon.cgi?Search_Arg=HB172.M6247+1995&Search_Code=CALL_&CNT=5



Chapter 18

Coreflexive relation

In mathematics, a coreflexive relation is a binary relation that is a subset of the identity relation.[1] Thus if a is relatedto b (aRb) then a is equal to b (a = b), but if c is equal to d (c = d) it does not necessarily hold that c is related to d(cRd).In mathematical notation, this is:

∀a, b ∈ X, aRb⇒ a = b.

The identity relation is coreflexive by definition. Any relation that is coreflexive is thus a subset of the identity relation.For example, consider the relation R as “equal to and odd”. Over the set of positive integers, the relationship R holdsover the pairs {(1, 1), (3, 3), ...} but does not hold over {(2, 2), (4, 4), ...}.

18.1 Notes[1] Fonseca de Oliveira, J. N., & Pereira Cunha Rodrigues, C. D. J. (2004). Transposing Relations: From Maybe Functions

to Hash Tables. In Mathematics of Program Construction (p. 337).

64



https://en.wikipedia.org/wiki/Identity_relation



Chapter 19

Demonic composition

In mathematics, demonic composition is an operation on binary relations that is somewhat comparable to ordinarycomposition of relations but is robust to refinement of the relations into (partial) functions or injective relations.Unlike ordinary composition of relations, demonic composition is not associative.

19.1 Definition

Suppose R is a binary relation between X and Y and S is a relation between Y and Z. Their right demonic compositionR ;→ S is a relation between X and Z. Its graph is defined as

{(x, z) | x (S ◦R) z ∧ ∀y ∈ Y (xR y ⇒ y S z)}.

Conversely, their left demonic composition R ;← S is defined by

{(x, z) | x (S ◦R) z ∧ ∀y ∈ Y (y S z ⇒ xR y)}.

19.2 References• Backhouse, Roland; van der Woude, Jaap (1993), “Demonic operators and monotype factors”, Mathematical

Structures in Computer Science 3 (4): 417–433, doi:10.1017/S096012950000030X, MR 1249420.

65






https://dx.doi.org/10.1017%252FS096012950000030X



Chapter 20

Dense order

In mathematics, a partial order < on a set X is said to be dense if, for all x and y in X for which x < y, there is a z inX such that x < z < y.

20.1 Example

The rational numbers with the ordinary ordering are a densely ordered set in this sense, as are the real numbers. Onthe other hand, the ordinary ordering on the integers is not dense.

20.2 Generalizations

Any binary relation R is said to be dense if, for all R-related x and y, there is a z such that x and z and also z and y areR-related. Formally:

∀x ∀y xRy ⇒ (∃z xRz ∧ zRy).

Every reflexive relation is dense. A strict partial order < is a dense order iff < is a dense relation.

20.3 See also• Dense set

• Dense-in-itself

• Kripke semantics

20.4 References• David Harel, Dexter Kozen, Jerzy Tiuryn, Dynamic logic, MIT Press, 2000, ISBN 0-262-08289-6, p. 6ff

66



https://en.wikipedia.org/wiki/Rational_number







https://en.wikipedia.org/wiki/Dense_set

https://en.wikipedia.org/wiki/Dense-in-itself

https://en.wikipedia.org/wiki/Kripke_semantics

https://en.wikipedia.org/wiki/David_Harel

https://en.wikipedia.org/wiki/Dexter_Kozen


Chapter 21

Dependence relation

Not to be confused with Dependency relation, which is a binary relation that is symmetric and reflexive.

In mathematics, a dependence relation is a binary relation which generalizes the relation of linear dependence.LetX be a set. A (binary) relation ◁ between an element a ofX and a subset S ofX is called a dependence relation,written a ◁ S , if it satisfies the following properties:

• if a ∈ S , then a ◁ S ;

• if a ◁ S , then there is a finite subset S0 of S , such that a ◁ S0 ;

• if T is a subset of X such that b ∈ S implies b ◁ T , then a ◁ S implies a ◁ T ;

• if a ◁ S but a ̸◁S − {b} for some b ∈ S , then b ◁ (S − {b}) ∪ {a} .

Given a dependence relation ◁ onX , a subset S ofX is said to be independent if a ̸◁S−{a} for all a ∈ S. If S ⊆ T, then S is said to span T if t ◁ S for every t ∈ T. S is said to be a basis of X if S is independent and S spans X.

Remark. If X is a non-empty set with a dependence relation ◁ , then X always has a basis with respect to ◁.Furthermore, any two bases of X have the same cardinality.

21.1 Examples• Let V be a vector space over a field F. The relation ◁ , defined by υ ◁ S if υ is in the subspace spanned by S ,

is a dependence relation. This is equivalent to the definition of linear dependence.

• LetK be a field extension of F.Define ◁ by α◁S if α is algebraic over F (S). Then ◁ is a dependence relation.This is equivalent to the definition of algebraic dependence.

21.2 See also• matroid

This article incorporates material from Dependence relation on PlanetMath, which is licensed under the Creative Com-mons Attribution/Share-Alike License.

67




https://en.wikipedia.org/wiki/Linear_dependence







https://en.wikipedia.org/wiki/Linear_subspace


https://en.wikipedia.org/wiki/Linear_independence

https://en.wikipedia.org/wiki/Field_extension

https://en.wikipedia.org/wiki/Algebraic_element

https://en.wikipedia.org/wiki/Algebraic_dependence

https://en.wikipedia.org/wiki/Matroid

http://planetmath.org/node/35792


https://en.wikipedia.org/wiki/Wikipedia:CC-BY-SA

https://en.wikipedia.org/wiki/Wikipedia:CC-BY-SA

Chapter 22

Dependency relation

For other uses, see Dependency (disambiguation).Not to be confused with Dependence relation, which is a generalization of the concept of linear dependence amongmembers of a vector space.

In mathematics and computer science, a dependency relation is a binary relation that is finite, symmetric, andreflexive; i.e. a finite tolerance relation. That is, it is a finite set of ordered pairs D , such that

• If (a, b) ∈ D then (b, a) ∈ D (symmetric)• If a is an element of the set on which the relation is defined, then (a, a) ∈ D (reflexive)

In general, dependency relations are not transitive; thus, they generalize the notion of an equivalence relation bydiscarding transitivity.Let Σ denote the alphabet of all the letters of D . Then the independency induced by D is the binary relation I

I = Σ× Σ \D

That is, the independency is the set of all ordered pairs that are not in D . The independency is symmetric andirreflexive.The pairs (Σ, D) and (Σ, I) , or the triple (Σ, D, I) (with I induced by D ) are sometimes called the concurrentalphabet or the reliance alphabet.The pairs of letters in an independency relation induce an equivalence relation on the free monoid of all possiblestrings of finite length. The elements of the equivalence classes induced by the independency are called traces, andare studied in trace theory.

22.1 Examples

Consider the alphabet Σ = {a, b, c} . A possible dependency relation is

D = {a, b} × {a, b} ∪ {a, c} × {a, c}= {a, b}2 ∪ {a, c}2

= {(a, b), (b, a), (a, c), (c, a), (a, a), (b, b), (c, c)}

The corresponding independency is

ID = {(b, c) , (c, b)}

Therefore, the letters b, c commute, or are independent of one another.

68

https://en.wikipedia.org/wiki/Dependency_(disambiguation)

https://en.wikipedia.org/wiki/Dependence_relation






https://en.wikipedia.org/wiki/Tolerance_relation




https://en.wikipedia.org/wiki/Alphabet_(computer_science)

https://en.wikipedia.org/wiki/Free_monoid


https://en.wikipedia.org/wiki/Trace_monoid

https://en.wikipedia.org/wiki/Trace_theory

22.1. EXAMPLES 69

Aa

b

c

Chapter 23






70









































23.1.2 Example































23.3. RELATIONS OVER A SET 73





23.2.1 Difunctional



















































































23.4.1 Complement








23.4.2 Restriction




































































23.7. EXAMPLES OF COMMON BINARY RELATIONS 77









23.8 See also


• Hasse diagram



• Order theory


















































































https://dx.doi.org/10.1007%252F978-3-540-74788-8_18




https://dx.doi.org/10.1007%252F978-3-540-27778-1_15







https://dx.doi.org/10.1007%252F978-3-662-44124-4_7






















23.11. EXTERNAL LINKS 79


55608-010-4







Chapter 24

Directed set

In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexiveand transitive binary relation ≤ (that is, a preorder), with the additional property that every pair of elements has anupper bound.[1] In other words, for any a and b in A there must exist a c in A with a ≤ c and b ≤ c.The notion defined above is sometimes called an upward directed set. A downward directed set is definedanalogously,[2] meaning when every doubleton is bounded below.[3] Some authors (and this article) assume that adirected set is directed upward, unless otherwise stated. Beware that other authors call a set directed if and only if itis directed both upward and downward.[4]

Directed sets are a generalization of nonempty totally ordered sets, that is, all totally ordered sets are directed sets(contrast partially ordered sets which need not be directed). Join semilattices (which are partially ordered sets) aredirected sets as well, but not conversely. Likewise, lattices are directed sets both upward and downward.In topology, directed sets are used to define nets, which generalize sequences and unite the various notions of limitused in analysis. Directed sets also give rise to direct limits in abstract algebra and (more generally) category theory.

24.1 Equivalent definition

In addition to the definition above, there is an equivalent definition. A directed set is a set A with a preorder suchthat every finite subset of A has an upper bound. In this definition, the existence of an upper bound of the emptysubset implies that A is nonempty.

24.2 Examples

Examples of directed sets include:

• The set of natural numbers N with the ordinary order ≤ is a directed set (and so is every totally ordered set).

• Let D1 and D2 be directed sets. Then the Cartesian product set D1 × D2 can be made into a directed set bydefining (n1, n2) ≤ (m1, m2) if and only if n1 ≤ m1 and n2 ≤ m2. In analogy to the product order this is theproduct direction on the Cartesian product.

• It follows from previous example that the set N × N of pairs of natural numbers can be made into a directedset by defining (n0, n1) ≤ (m0, m1) if and only if n0 ≤ m0 and n1 ≤ m1.

• If x0 is a real number, we can turn the set R − {x0} into a directed set by writing a ≤ b if and only if|a − x0| ≥ |b − x0|. We then say that the reals have been directed towards x0. This is an example of a directedset that is not ordered (neither totally nor partially).

• A (trivial) example of a partially ordered set that is not directed is the set {a, b}, in which the only orderrelations are a ≤ a and b ≤ b. A less trivial example is like the previous example of the “reals directed towardsx0" but in which the ordering rule only applies to pairs of elements on the same side of x0.

80








https://en.wikipedia.org/wiki/Totally_ordered_set

https://en.wikipedia.org/wiki/Partially_ordered_sets

https://en.wikipedia.org/wiki/Semilattice


https://en.wikipedia.org/wiki/Topology

https://en.wikipedia.org/wiki/Net_(topology)

https://en.wikipedia.org/wiki/Sequence

https://en.wikipedia.org/wiki/Limit_(mathematics)

https://en.wikipedia.org/wiki/Mathematical_analysis

https://en.wikipedia.org/wiki/Direct_limit







https://en.wikipedia.org/wiki/Product_order



24.3. CONTRAST WITH SEMILATTICES 81

• If T is a topological space and x0 is a point in T, we turn the set of all neighbourhoods of x0 into a directed setby writing U ≤ V if and only if U contains V.

• For every U: U ≤ U; since U contains itself.• For every U,V,W : if U ≤ V and V ≤ W, then U ≤ W; since if U contains V and V contains W then U

contains W.• For every U, V: there exists the set U ∩ V such that U ≤ U ∩ V and V ≤ U ∩ V; since both U and V

contain U ∩ V.

• In a poset P, every lower closure of an element, i.e. every subset of the form {a| a in P, a ≤x} where x is afixed element from P, is directed.

24.3 Contrast with semilattices

Witness

Directed sets are a more general concept than (join) semilattices: every join semilattice is a directed set, as the joinor least upper bound of two elements is the desired c. The converse does not hold however, witness the directed set


https://en.wikipedia.org/wiki/Topological_neighbourhood

https://en.wikipedia.org/wiki/Poset


82 CHAPTER 24. DIRECTED SET

{1000,0001,1101,1011,1111} ordered bitwise (e.g. 1000 ≤ 1011 holds, but 0001 ≤ 1000 does not, since in the lastbit 1 > 0), where {1000,0001} has three upper bounds but no least upper bound, cf. picture. (Also note that without1111, the set is not directed.)

24.4 Directed subsets

The order relation in a directed sets is not required to be antisymmetric, and therefore directed sets are not alwayspartial orders. However, the term directed set is also used frequently in the context of posets. In this setting, a subsetA of a partially ordered set (P,≤) is called a directed subset if it is a directed set according to the same partial order:in other words, it is not the empty set, and every pair of elements has an upper bound. Here the order relation on theelements of A is inherited from P; for this reason, reflexivity and transitivity need not be required explicitly.A directed subset of a poset is not required to be downward closed; a subset of a poset is directed if and only if itsdownward closure is an ideal. While the definition of a directed set is for an “upward-directed” set (every pair ofelements has an upper bound), it is also possible to define a downward-directed set in which every pair of elementshas a common lower bound. A subset of a poset is downward-directed if and only if its upper closure is a filter.Directed subsets are used in domain theory, which studies directed complete partial orders.[5] These are posets inwhich every upward-directed set is required to have a least upper bound. In this context, directed subsets againprovide a generalization of convergent sequences.

24.5 See also• Filtered category

• Centered set

• Linked set

24.6 Notes[1] Kelley, p. 65.

[2] Robert S. Borden (1988). A Course in Advanced Calculus. Courier Corporation. p. 20. ISBN 978-0-486-15038-3.

[3] Arlen Brown; Carl Pearcy (1995). An Introduction to Analysis. Springer. p. 13. ISBN 978-1-4612-0787-0.

[4] Siegfried Carl; Seppo Heikkilä (2010). Fixed Point Theory in Ordered Sets and Applications: From Differential and IntegralEquations to Game Theory. Springer. p. 77. ISBN 978-1-4419-7585-0.

[5] Gierz, p. 2.

24.7 References• J. L. Kelley (1955), General Topology.

• Gierz, Hofmann, Keimel, et al. (2003), Continuous Lattices and Domains, Cambridge University Press. ISBN0-521-80338-1.

https://en.wikipedia.org/wiki/Coordinatewise_order




https://en.wikipedia.org/wiki/Lower_set

https://en.wikipedia.org/wiki/Ideal_(order_theory)

https://en.wikipedia.org/wiki/Filter_(mathematics)

https://en.wikipedia.org/wiki/Domain_theory

https://en.wikipedia.org/wiki/Directed_complete_partial_order

https://en.wikipedia.org/wiki/Least_upper_bound

https://en.wikipedia.org/wiki/Filtered_category

https://en.wikipedia.org/wiki/Centered_set

https://en.wikipedia.org/wiki/Linked_set









Chapter 25

Equality (mathematics)

In mathematics, equality is a relationship between two quantities or, more generally two mathematical expressions,asserting that the quantities have the same value or that the expressions represent the same mathematical object. Theequality between A and B is written A = B, and pronounced A equals B. The symbol "=" is called an "equals sign".

25.1 Etymology

The etymology of the word is from the Latin aequālis (“equal”, “like”, “comparable”, “similar”) from aequus (“equal”,“level”, “fair”, “just”).

25.2 Types of equalities

25.2.1 Identities

Main article: Identity (mathematics)

When A and B may be viewed as functions of some variables, then A = B means that A and B define the same function.Such an equality of functions is sometimes called an identity. An example is (x + 1)2 = x2 + 2x + 1.

25.2.2 Equalities as predicates

When A and B are not fully specified or depend on some variables, equality is a proposition, which may be truefor some values and false for some other values. Equality is a binary relation, or, in other words, a two-argumentspredicate, which may produce a truth value (false or true) from its arguments. In computer programming, its com-putation from two expressions is known as comparison.

25.2.3 Congruences

In some cases, one may consider as equal two mathematical objects that are only equivalent for the properties thatare considered. This is, in particular the case in geometry, where two geometric shapes are said equal when one maybe moved to coincide with the other. The word congruence is also used for this kind of equality.

25.2.4 Equations

An equation is the problem of finding values of some variables, called unknowns, for which the specified equalityis true. Equation may also refer to an equality relation that is satisfied only for the values of the variables that oneis interested on. For example x2 + y2 = 1 is the equation of the unit circle. There is no standard notation that

83


https://en.wikipedia.org/wiki/Mathematical_expression

https://en.wikipedia.org/wiki/Mathematical_object

https://en.wikipedia.org/wiki/Equals_sign

https://en.wikipedia.org/wiki/Etymology

https://en.wiktionary.org/wiki/aequalis#Latin

https://en.wiktionary.org/wiki/aequus#Latin

https://en.wikipedia.org/wiki/Identity_(mathematics)


https://en.wikipedia.org/wiki/Identity_(mathematics)

https://en.wikipedia.org/wiki/Variable_(mathematics)

https://en.wikipedia.org/wiki/Proposition_(mathematics)


https://en.wikipedia.org/wiki/Predicate_(mathematical_logic)

https://en.wikipedia.org/wiki/Truth_value

https://en.wikipedia.org/wiki/Computer_programming

https://en.wikipedia.org/wiki/Relational_operator


https://en.wikipedia.org/wiki/Geometric_shape

https://en.wikipedia.org/wiki/Equation

https://en.wikipedia.org/wiki/Unit_circle

84 CHAPTER 25. EQUALITY (MATHEMATICS)

distinguishes an equation from an identity or other use of the equality relation: a reader has to guess an appropriateinterpretation from the semantic of expressions and the context.

25.2.5 Equivalence relations

Main article: Equivalence relation

Viewed as a relation, equality is the archetype of the more general concept of an equivalence relation on a set:those binary relations that are reflexive, symmetric, and transitive. The identity relation is an equivalence relation.Conversely, let R be an equivalence relation, and let us denote by xR the equivalence class of x, consisting of allelements z such that x R z. Then the relation x R y is equivalent with the equality xR = yR. It follows that equalityis the smallest equivalence relation on any set S, in the sense that it is the relation that has the smallest equivalenceclasses (every class is reduced to a single element).

25.3 Logical formalizations of equality

There are several formalizations of the notion of equality in mathematical logic, usually by means of axioms, such asthe first few Peano axioms, or the axiom of extensionality in ZF set theory.For example, Azriel Lévy gives as the five axioms for equality, first the three properties of an equivalence relation,and these two:

x = y ∧ x ∈ z ⇒ y ∈ z, andx = y ∧ z ∈ x ⇒ z ∈ y.[1]

These extra two conditions allow substitution of equal quantities into complex expressions.There are also some logic systems that do not have any notion of equality. This reflects the undecidability of theequality of two real numbers defined by formulas involving the integers, the basic arithmetic operations, the logarithmand the exponential function. In other words, there cannot exist any algorithm for deciding such an equality.

25.4 Logical formulations

Equality is always defined such that things that are equal have all and only the same properties. Some people defineequality as congruence. Often equality is just defined as identity.A stronger sense of equality is obtained if some form of Leibniz’s law is added as an axiom; the assertion of this axiomrules out “bare particulars”—things that have all and only the same properties but are not equal to each other—whichare possible in some logical formalisms. The axiom states that two things are equal if they have all and only the sameproperties. Formally:

Given any x and y, x = y if, given any predicate P, P(x) if and only if P(y).

In this law, the connective “if and only if” can be weakened to “if"; the modified law is equivalent to the original.Instead of considering Leibniz’s law as an axiom, it can also be taken as the definition of equality. The property ofbeing an equivalence relation, as well as the properties given below, can then be proved: they become theorems. Ifa=b, then a can replace b and b can replace a.

25.5 Some basic logical properties of equality

The substitution property states:

• For any quantities a and b and any expression F(x), if a = b, then F(a) = F(b) (if both sides make sense, i.e.are well-formed).







https://en.wikipedia.org/wiki/Peano_axioms

https://en.wikipedia.org/wiki/Axiom_of_extensionality

https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory

https://en.wikipedia.org/wiki/Azriel_L%C3%A9vy


https://en.wikipedia.org/wiki/Substitution_(algebra)


https://en.wikipedia.org/wiki/Undecidable_problem



https://en.wikipedia.org/wiki/Arithmetic_operation

https://en.wikipedia.org/wiki/Logarithm


https://en.wikipedia.org/wiki/Algorithm

https://en.wikipedia.org/wiki/Identity_(philosophy)

https://en.wikipedia.org/wiki/Identity_of_indiscernibles

https://en.wikipedia.org/wiki/Axiom

https://en.wikipedia.org/wiki/Property_(philosophy)

https://en.wikipedia.org/wiki/Given_any


https://en.wikipedia.org/wiki/Predicate_(mathematics)


https://en.wikipedia.org/wiki/Theorem

https://en.wikipedia.org/wiki/For_any


https://en.wikipedia.org/wiki/Well-formed_formula

25.6. RELATION WITH EQUIVALENCE AND ISOMORPHISM 85

In first-order logic, this is a schema, since we can't quantify over expressions like F (which would be a functionalpredicate).Some specific examples of this are:

• For any real numbers a, b, and c, if a = b, then a + c = b + c (here F(x) is x + c);

• For any real numbers a, b, and c, if a = b, then a − c = b − c (here F(x) is x − c);

• For any real numbers a, b, and c, if a = b, then ac = bc (here F(x) is xc);

• For any real numbers a, b, and c, if a = b and c is not zero, then a/c = b/c (here F(x) is x/c).

The reflexive property states:

For any quantity a, a = a.

This property is generally used in mathematical proofs as an intermediate step.The symmetric property states:

• For any quantities a and b, if a = b, then b = a.

The transitive property states:

• For any quantities a, b, and c, if a = b and b = c, then a = c.

The binary relation "is approximately equal" between real numbers or other things, even if more precisely defined,is not transitive (it may seem so at first sight, but many small differences can add up to something big). However,equality almost everywhere is transitive.Although the symmetric and transitive properties are often seen as fundamental, they can be proved, if the substitutionand reflexive properties are assumed instead.

25.6 Relation with equivalence and isomorphism

See also: Equivalence relation and Isomorphism

In some contexts, equality is sharply distinguished from equivalence or isomorphism.[2] For example, one may distin-guish fractions from rational numbers, the latter being equivalence classes of fractions: the fractions 1/2 and 2/4 aredistinct as fractions, as different strings of symbols, but they “represent” the same rational number, the same pointon a number line. This distinction gives rise to the notion of a quotient set.Similarly, the sets

{A,B,C} and {1, 2, 3}

are not equal sets – the first consists of letters, while the second consists of numbers – but they are both sets of threeelements, and thus isomorphic, meaning that there is a bijection between them, for example

A 7→ 1,B 7→ 2,C 7→ 3.

However, there are other choices of isomorphism, such as

A 7→ 3,B 7→ 2,C 7→ 1,

and these sets cannot be identified without making such a choice – any statement that identifies them “dependson choice of identification”. This distinction, between equality and isomorphism, is of fundamental importance incategory theory, and is one motivation for the development of category theory.


https://en.wikipedia.org/wiki/Schema_(logic)

https://en.wikipedia.org/wiki/Functional_predicate

https://en.wikipedia.org/wiki/Functional_predicate





https://en.wikipedia.org/wiki/Division_by_zero

https://en.wikipedia.org/wiki/0_(number)


https://en.wikipedia.org/wiki/Mathematical_proof





https://en.wikipedia.org/wiki/And_(logic)


https://en.wikipedia.org/wiki/Approximation


https://en.wikipedia.org/wiki/Difference_(mathematics)

https://en.wikipedia.org/wiki/Almost_everywhere





https://en.wikipedia.org/wiki/Fraction_(mathematics)


https://en.wikipedia.org/wiki/Quotient_set


https://en.wikipedia.org/wiki/Isomorphism#Relation_with_equality


86 CHAPTER 25. EQUALITY (MATHEMATICS)

25.7 See also• Equals sign

• Inequality

• Logical equality

• Extensionality

25.8 References[1] Azriel Lévy (1979) Basic Set Theory, page 358, Springer-Verlag

[2] (Mazur 2007)

• Mazur, Barry (12 June 2007), When is one thing equal to some other thing? (PDF)

• Mac Lane, Saunders; Garrett Birkhoff (1967). Algebra. American Mathematical Society.

https://en.wikipedia.org/wiki/Equals_sign


https://en.wikipedia.org/wiki/Logical_equality

https://en.wikipedia.org/wiki/Extensionality

https://en.wikipedia.org/wiki/Azriel_L%C3%A9vy

https://en.wikipedia.org/wiki/Equality_(mathematics)#CITEREFMazur2007

https://en.wikipedia.org/wiki/Barry_Mazur

http://www.math.harvard.edu/~mazur/preprints/when_is_one.pdf

https://en.wikipedia.org/wiki/Saunders_Mac_Lane

https://en.wikipedia.org/wiki/Garrett_Birkhoff

Chapter 26

Equipollence (geometry)

In Euclidean geometry, equipollence is a binary relation between directed line segments. A line segment AB frompoint A to point B has the opposite direction to line segment BA. Two directed line segments are equipollent whenthey have the same length and direction.The concept of equipollent line segments was advanced by Giusto Bellavitis in 1835. Subsequently the term vectorwas adopted for a class of equipollent line segments. Bellavitis’s use of the idea of a relation to compare differentbut similar objects has become a common mathematical technique, particularly in the use of equivalence relations.Bellavitis used a special notation for the equipollence of segments AB and CD:

AB ≏ CD.

The following passages, translated by Michael J. Crowe, show the anticipation that Bellavitis had of vector concepts:

Equipollences continue to hold when one substitutes for the lines in them, other lines which are respec-tively equipollent to them, however they may be situated in space. From this it can be understood howany number and any kind of lines may be summed, and that in whatever order these lines are taken, thesame equipollent-sum will be obtained...

In equipollences, just as in equations, a line may be transferred from one side to the other, provided thatthe sign is changed...

Thus oppositely directed segments are negatives of each other: AB +BA ≏ 0.

The equipollence AB ≏ n.CD, where n stands for a positive number, indicates that AB is both parallelto and has the same direction as CD, and that their lengths have the relation expressed by AB = n.CD .

26.1 References

• Giusto Bellavitis (1835) “Saggio di applicasioni di un nuovo metodo di Geometria Analitica (Calculo delleequipollenze)", Annali delle Scienze del Regno Lombardo-Veneto, Padova 5: 244–59.

• Giusto Bellavitis (1854) Sposizione del Metodo della Equipollenze, link from Google Books.

• Michael J. Crowe (1967) A History of Vector Analysis, “Giusto Bellavitis and His Calculus of Equipollences”,pp 52–4, University of Notre Dame Press.

• Lena L. Severance (1930) The Theory of Equipollences; Method of Analytical Geometry of Sig. Bellavitis,link from HathiTrust.

87

https://en.wikipedia.org/wiki/Euclidean_geometry


https://en.wikipedia.org/wiki/Line_segment#Directed_line_segment

https://en.wikipedia.org/wiki/Giusto_Bellavitis



https://en.wikipedia.org/wiki/Euclidean_vector

https://en.wikipedia.org/wiki/Giusto_Bellavitis

http://books.google.ca/books?id=sHK6JCmktt0C

https://en.wikipedia.org/wiki/Google_Books

https://en.wikipedia.org/wiki/A_History_of_Vector_Analysis

https://en.wikipedia.org/wiki/University_of_Notre_Dame_Press

http://babel.hathitrust.org/cgi/pt?id=mdp.39015069379678;view=1up;seq=15

88 CHAPTER 26. EQUIPOLLENCE (GEOMETRY)

26.2 External links• Axiomatic definition of equipollence

http://sites.google.com/site/contributionsingeometry

Chapter 27

Equivalence class

This article is about equivalency in mathematics. For equivalency in music, see equivalence class (music).In mathematics, when a set has an equivalence relation defined on its elements, there is a natural grouping of

Congruence is an example of an equivalence relation. The two triangles on the left are congruent, while the third and fourth trianglesare not congruent to any other triangle. Thus, the first two triangles are in the same equivalence class, while the third and fourthtriangles are in their own equivalence class.

elements that are related to one another, forming what are called equivalence classes. Notationally, given a set Xand an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X whichare equivalent to a. It follows from the definition of the equivalence relations that the equivalence classes form apartition of X. The set of equivalence classes is sometimes called the quotient set or the quotient space of X by ~and is denoted by X / ~.When X has some structure, and the equivalence relation is defined with some connection to this structure, thequotient set often inherits some related structure. Examples include quotient spaces in linear algebra, quotient spacesin topology, quotient groups, homogeneous spaces, quotient rings, quotient monoids, and quotient categories.

27.1 Notation and formal definition

An equivalence relation is a binary relation ~ satisfying three properties:[1]

• For every element a in X, a ~ a (reflexivity),

• For every two elements a and b in X, if a ~ b, then b ~ a (symmetry)

• For every three elements a, b, and c in X, if a ~ b and b ~ c, then a ~ c (transitivity).

89

https://en.wikipedia.org/wiki/Equivalence_class_(music)






https://en.wikipedia.org/wiki/Quotient_space_(linear_algebra)

https://en.wikipedia.org/wiki/Quotient_space_(topology)



https://en.wikipedia.org/wiki/Homogeneous_space


https://en.wikipedia.org/wiki/Quotient_monoid

https://en.wikipedia.org/wiki/Quotient_category





90 CHAPTER 27. EQUIVALENCE CLASS

The equivalence class of an element a is denoted [a] and is defined as the set

[a] = {x ∈ X | a ∼ x}

of elements that are related to a by ~. An alternative notation [a]R can be used to denote the equivalence class of theelement a, specifically with respect to the equivalence relation R. This is said to be the R-equivalence class of a.The set of all equivalence classes in X with respect to an equivalence relation R is denoted as X/R and called XmoduloR (or the quotient set of X by R).[2] The surjective map x 7→ [x] from X onto X/R, which maps each element to itsequivalence class, is called the canonical surjection or the canonical projection map.When an element is chosen (often implicitly) in each equivalence class, this defines an injective map called a section. Ifthis section is denoted by s, one has [s(c)] = c for every equivalence class c. The element s(c) is called a representativeof c. Any element of a class may be chosen as a representative of the class, by choosing the section appropriately.Sometimes, there is a section that is more “natural” than the other ones. In this case, the representatives are calledcanonical representatives. For example, in modular arithmetic, consider the equivalence relation on the integersdefined by a ~ b if a − b is a multiple of a given integer n, called the modulus. Each class contains a unique non-negative integer smaller than n, and these integers are the canonical representatives. The class and its representativeare more or less identified, as is witnessed by the fact that the notation a mod n may denote either the class or itscanonical representative (which is the remainder of the division of a by n).

27.2 Examples• If X is the set of all cars, and ~ is the equivalence relation “has the same color as.” then one particular equivalence

class consists of all green cars. X/~ could be naturally identified with the set of all car colors (cardinality ofX/~ would be the number of all car colors).

• Let X be the set of all rectangles in a plane, and ~ the equivalence relation “has the same area as”. For eachpositive real number A there will be an equivalence class of all the rectangles that have area A.[3]

• Consider the modulo 2 equivalence relation on the set Z of integers: x ~ y if and only if their difference x − yis an even number. This relation gives rise to exactly two equivalence classes: one class consisting of all evennumbers, and the other consisting of all odd numbers. Under this relation [7], [9], and [1] all represent thesame element of Z/~.[4]

• Let X be the set of ordered pairs of integers (a,b) with b not zero, and define an equivalence relation ~ on Xaccording to which (a,b) ~ (c,d) if and only if ad = bc. Then the equivalence class of the pair (a,b) can beidentified with the rational number a/b, and this equivalence relation and its equivalence classes can be used togive a formal definition of the set of rational numbers.[5] The same construction can be generalized to the fieldof fractions of any integral domain.

• If X consists of all the lines in, say the Euclidean plane, and L ~ M means that L and M are parallel lines, thenthe set of lines that are parallel to each other form an equivalence class as long as a line is considered parallelto itself. In this situation, each equivalence class determines a point at infinity.

27.3 Properties

Every element x of X is a member of the equivalence class [x]. Every two equivalence classes [x] and [y] are eitherequal or disjoint. Therefore, the set of all equivalence classes of X forms a partition of X: every element of X belongsto one and only one equivalence class.[6] Conversely every partition of X comes from an equivalence relation in thisway, according to which x ~ y if and only if x and y belong to the same set of the partition.[7]

It follows from the properties of an equivalence relation that

x ~ y if and only if [x] = [y].

In other words, if ~ is an equivalence relation on a set X, and x and y are two elements of X, then these statementsare equivalent:

https://en.wikipedia.org/wiki/Surjective_map

https://en.wikipedia.org/wiki/Injective_map

https://en.wikipedia.org/wiki/Section_(category_theory)

https://en.wikipedia.org/wiki/Canonical_form


https://en.wikipedia.org/wiki/Remainder

https://en.wikipedia.org/wiki/Euclidean_division




https://en.wikipedia.org/wiki/Even_number


https://en.wikipedia.org/wiki/Field_of_fractions

https://en.wikipedia.org/wiki/Field_of_fractions

https://en.wikipedia.org/wiki/Integral_domain

https://en.wikipedia.org/wiki/Euclidean_plane

https://en.wikipedia.org/wiki/Parallel_lines

https://en.wikipedia.org/wiki/Parallel_(geometry)#Reflexive_variant

https://en.wikipedia.org/wiki/Parallel_(geometry)#Reflexive_variant

https://en.wikipedia.org/wiki/Point_at_infinity

https://en.wikipedia.org/wiki/Disjoint_sets


27.4. GRAPHICAL REPRESENTATION 91

• x ∼ y

• [x] = [y]

• [x] ∩ [y] ̸= ∅.

27.4 Graphical representation

Any binary relation can be represented by a directed graph and symmetric ones, such as equivalence relations, byundirected graphs. If ~ is an equivalence relation on a set X, let the vertices of the graph be the elements of X andjoin vertices s and t if and only if s ~ t. The equivalence classes are represented in this graph by the maximal cliquesforming the connected components of the graph.[8]

27.5 Invariants

If ~ is an equivalence relation on X, and P(x) is a property of elements of X such that whenever x ~ y, P(x) is true ifP(y) is true, then the property P is said to be an invariant of ~, or well-defined under the relation ~.A frequent particular case occurs when f is a function from X to another set Y; if f(x1) = f(x2) whenever x1 ~ x2,then f is said to be a morphism for ~, a class invariant under ~, or simply invariant under ~. This occurs, e.g. in thecharacter theory of finite groups. Some authors use “compatible with ~" or just “respects ~" instead of “invariantunder ~".Any function f : X → Y itself defines an equivalence relation on X according to which x1 ~ x2 if and only if f(x1)= f(x2). The equivalence class of x is the set of all elements in X which get mapped to f(x), i.e. the class [x] is theinverse image of f(x). This equivalence relation is known as the kernel of f.More generally, a function may map equivalent arguments (under an equivalence relation ~X on X) to equivalentvalues (under an equivalence relation ~Y on Y). Such a function is known as a morphism from ~X to ~Y .

27.6 Quotient space in topology

In topology, a quotient space is a topological space formed on the set of equivalence classes of an equivalence relationon a topological space using the original space’s topology to create the topology on the set of equivalence classes.In abstract algebra, congruence relations on the underlying set of an algebra allow the algebra to induce an algebraon the equivalence classes of the relation, called a quotient algebra. In linear algebra, a quotient space is a vectorspace formed by taking a quotient group where the quotient homomorphism is a linear map. By extension, in abstractalgebra, the term quotient space may be used for quotient modules, quotient rings, quotient groups, or any quotientalgebra. However, the use of the term for the more general cases can as often be by analogy with the orbits of a groupaction.The orbits of a group action on a set may be called the quotient space of the action on the set, particularly when theorbits of the group action are the right cosets of a subgroup of a group, which arise from the action of the subgroupon the group by left translations, or respectively the left cosets as orbits under right translation.A normal subgroup of a topological group, acting on the group by translation action, is a quotient space in the sensesof topology, abstract algebra, and group actions simultaneously.Although the term can be used for any equivalence relation’s set of equivalence classes, possibly with further structure,the intent of using the term is generally to compare that type of equivalence relation on a set X either to an equivalencerelation that induces some structure on the set of equivalence classes from a structure of the same kind on X, or to theorbits of a group action. Both the sense of a structure preserved by an equivalence relation and the study of invariantsunder group actions lead to the definition of invariants of equivalence relations given above.

27.7 See also• Equivalence partitioning, a method for devising test sets in software testing based on dividing the possible



https://en.wikipedia.org/wiki/Graph_(mathematics)

https://en.wikipedia.org/wiki/Glossary_of_graph_theory

https://en.wikipedia.org/wiki/Connected_component_(graph_theory)

https://en.wikipedia.org/wiki/Invariant_(mathematics)


https://en.wikipedia.org/wiki/Morphism


https://en.wikipedia.org/wiki/Inverse_image

https://en.wikipedia.org/wiki/Kernel_of_a_function





https://en.wikipedia.org/wiki/Congruence_relation



https://en.wikipedia.org/wiki/Quotient_space_(linear_algebra)


https://en.wikipedia.org/wiki/Linear_map

https://en.wikipedia.org/wiki/Quotient_module





https://en.wikipedia.org/wiki/Group_action



https://en.wikipedia.org/wiki/Equivalence_class#Invariants

https://en.wikipedia.org/wiki/Equivalence_partitioning

https://en.wikipedia.org/wiki/Software_testing

92 CHAPTER 27. EQUIVALENCE CLASS

program inputs into equivalence classes according to the behavior of the program on those inputs

• Homogeneous space, the quotient space of Lie groups.

• Transversal (combinatorics)

27.8 Notes[1] Devlin 2004, p. 122

[2] Wolf 1998, p. 178

[3] Avelsgaard 1989, p. 127

[4] Devlin 2004, p. 123

[5] Maddox 2002, pp. 77–78

[6] Maddox 2002, p.74, Thm. 2.5.15

[7] Avelsgaard 1989, p.132, Thm. 3.16

[8] Devlin 2004, p. 123

27.9 References• Avelsgaard, Carol (1989), Foundations for Advanced Mathematics, Scott Foresman, ISBN 0-673-38152-8

• Devlin, Keith (2004), Sets, Functions, and Logic: An Introduction to Abstract Mathematics (3rd ed.), Chapman& Hall/ CRC Press, ISBN 978-1-58488-449-1

• Maddox, Randall B. (2002), Mathematical Thinking and Writing, Harcourt/ Academic Press, ISBN 0-12-464976-9

• Morash, Ronald P. (1987), Bridge to Abstract Mathematics, Random House, ISBN 0-394-35429-X

• Wolf, Robert S. (1998), Proof, Logic and Conjecture: A Mathematician’s Toolbox, Freeman, ISBN 978-0-7167-3050-7

27.10 Further reading

This material is basic and can be found in any text dealing with the fundamentals of proof technique, such as any ofthe following:

• Sundstrom (2003), Mathematical Reasoning: Writing and Proof, Prentice-Hall

• Smith; Eggen; St.Andre (2006), A Transition to Advanced Mathematics (6th Ed.), Thomson (Brooks/Cole)

• Schumacher, Carol (1996), Chapter Zero: Fundamental Notions of Abstract Mathematics, Addison-Wesley,ISBN 0-201-82653-4

• O'Leary (2003), The Structure of Proof: With Logic and Set Theory, Prentice-Hall

• Lay (2001), Analysis with an introduction to proof, Prentice Hall

• Gilbert; Vanstone (2005), An Introduction to Mathematical Thinking, Pearson Prentice-Hall

• Fletcher; Patty, Foundations of Higher Mathematics, PWS-Kent

• Iglewicz; Stoyle, An Introduction to Mathematical Reasoning, MacMillan

• D'Angelo; West (2000), Mathematical Thinking: Problem Solving and Proofs, Prentice Hall

https://en.wikipedia.org/wiki/Homogeneous_space

https://en.wikipedia.org/wiki/Lie_group

https://en.wikipedia.org/wiki/Transversal_(combinatorics)

https://en.wikipedia.org/wiki/Equivalence_class#CITEREFDevlin2004

https://en.wikipedia.org/wiki/Equivalence_class#CITEREFWolf1998

https://en.wikipedia.org/wiki/Equivalence_class#CITEREFAvelsgaard1989


https://en.wikipedia.org/wiki/Equivalence_class#CITEREFMaddox2002

https://en.wikipedia.org/wiki/Equivalence_class#CITEREFMaddox2002

https://en.wikipedia.org/wiki/Equivalence_class#CITEREFAvelsgaard1989
















27.10. FURTHER READING 93

• Cupillari, The Nuts and Bolts of Proofs, Wadsworth

• Bond, Introduction to Abstract Mathematics, Brooks/Cole

• Barnier; Feldman (2000), Introduction to Advanced Mathematics, Prentice Hall

• Ash, A Primer of Abstract Mathematics, MAA

Chapter 28

Equivalence relation

This article is about the mathematical concept. For the patent doctrine, see Doctrine of equivalents.In mathematics, an equivalence relation is the relation that holds between two elements if and only if they are

members of the same cell within a set that has been partitioned into cells such that every element of the set is amember of one and only one cell of the partition. The intersection of any two different cells is empty; the union ofall the cells equals the original set. These cells are formally called equivalence classes.

28.1 Notation

Although various notations are used throughout the literature to denote that two elements a and b of a set are equivalentwith respect to an equivalence relation R, the most common are "a ~ b" and "a ≡ b", which are used when R is theobvious relation being referenced, and variations of "a ~R b", "a ≡R b", or "aRb" otherwise.

28.2 Definition

A given binary relation ~ on a set X is said to be an equivalence relation if and only if it is reflexive, symmetric andtransitive. Equivalently, for all a, b and c in X:

• a ~ a. (Reflexivity)

• if a ~ b then b ~ a. (Symmetry)

• if a ~ b and b ~ c then a ~ c. (Transitivity)

X together with the relation ~ is called a setoid. The equivalence class of a under ~, denoted [a], is defined as[a] = {b ∈ X | a ∼ b} .

28.3 Examples

28.3.1 Simple example

Let the set {a, b, c} have the equivalence relation {(a, a), (b, b), (c, c), (b, c), (c, b)} . The following sets are equivalenceclasses of this relation:[a] = {a}, [b] = [c] = {b, c} .The set of all equivalence classes for this relation is {{a}, {b, c}} .

94

https://en.wikipedia.org/wiki/Doctrine_of_equivalents













https://en.wikipedia.org/wiki/Setoid




28.4. CONNECTIONS TO OTHER RELATIONS 95

28.3.2 Equivalence relations

The following are all equivalence relations:

• “Has the same birthday as” on the set of all people.

• “Is similar to” on the set of all triangles.

• “Is congruent to” on the set of all triangles.

• “Is congruent to, modulo n" on the integers.

• “Has the same image under a function" on the elements of the domain of the function.

• “Has the same absolute value” on the set of real numbers

• “Has the same cosine” on the set of all angles.

28.3.3 Relations that are not equivalences

• The relation "≥" between real numbers is reflexive and transitive, but not symmetric. For example, 7 ≥ 5 doesnot imply that 5 ≥ 7. It is, however, a partial order.

• The relation “has a common factor greater than 1 with” between natural numbers greater than 1, is reflexiveand symmetric, but not transitive. (Example: The natural numbers 2 and 6 have a common factor greater than1, and 6 and 3 have a common factor greater than 1, but 2 and 3 do not have a common factor greater than 1).

• The empty relation R on a non-empty set X (i.e. aRb is never true) is vacuously symmetric and transitive, butnot reflexive. (If X is also empty then R is reflexive.)

• The relation “is approximately equal to” between real numbers, even if more precisely defined, is not an equiv-alence relation, because although reflexive and symmetric, it is not transitive, since multiple small changes canaccumulate to become a big change. However, if the approximation is defined asymptotically, for example bysaying that two functions f and g are approximately equal near some point if the limit of f − g is 0 at that point,then this defines an equivalence relation.

28.4 Connections to other relations

• A partial order is a relation that is reflexive, antisymmetric, and transitive.

• Equality is both an equivalence relation and a partial order. Equality is also the only relation on a set thatis reflexive, symmetric and antisymmetric. In algebraic expressions, equal variables may be substituted forone another, a facility that is not available for equivalence related variables. The equivalence classes of anequivalence relation can substitute for one another, but not individuals within a class.

• A strict partial order is irreflexive, transitive, and asymmetric.

• A partial equivalence relation is transitive and symmetric. Transitive and symmetric imply reflexive if and onlyif for all a ∈ X, there exists a b ∈ X such that a ~ b.

• A reflexive and symmetric relation is a dependency relation, if finite, and a tolerance relation if infinite.

• A preorder is reflexive and transitive.

• A congruence relation is an equivalence relation whose domain X is also the underlying set for an algebraicstructure, and which respects the additional structure. In general, congruence relations play the role of kernelsof homomorphisms, and the quotient of a structure by a congruence relation can be formed. In many importantcases congruence relations have an alternative representation as substructures of the structure on which theyare defined. E.g. the congruence relations on groups correspond to the normal subgroups.

https://en.wikipedia.org/wiki/Similarity_(geometry)

https://en.wikipedia.org/wiki/Triangle_(geometry)


https://en.wikipedia.org/wiki/Triangle_(geometry)


https://en.wikipedia.org/wiki/Integers




https://en.wikipedia.org/wiki/Partially_ordered_set

https://en.wikipedia.org/wiki/Common_factor

https://en.wikipedia.org/wiki/Natural_numbers

https://en.wikipedia.org/wiki/Non-empty

https://en.wikipedia.org/wiki/Vacuously_true






https://en.wikipedia.org/wiki/Algebraic_expression

https://en.wikipedia.org/wiki/Substitution_(algebra)













https://en.wikipedia.org/wiki/Kernel_(algebra)


96 CHAPTER 28. EQUIVALENCE RELATION

28.5 Well-definedness under an equivalence relation

If ~ is an equivalence relation on X, and P(x) is a property of elements of X, such that whenever x ~ y, P(x) is true ifP(y) is true, then the property P is said to be well-defined or a class invariant under the relation ~.A frequent particular case occurs when f is a function from X to another set Y ; if x1 ~ x2 implies f(x1) = f(x2) thenf is said to be a morphism for ~, a class invariant under ~, or simply invariant under ~. This occurs, e.g. in thecharacter theory of finite groups. The latter case with the function f can be expressed by a commutative triangle. Seealso invariant. Some authors use “compatible with ~" or just “respects ~" instead of “invariant under ~".More generally, a function may map equivalent arguments (under an equivalence relation ~A) to equivalent values(under an equivalence relation ~B). Such a function is known as a morphism from ~A to ~B.

28.6 Equivalence class, quotient set, partition

Let a, b ∈ X . Some definitions:

28.6.1 Equivalence class

Main article: Equivalence class

A subset Y of X such that a ~ b holds for all a and b in Y, and never for a in Y and b outside Y, is called an equivalenceclass of X by ~. Let [a] := {x ∈ X | a ∼ x} denote the equivalence class to which a belongs. All elements of Xequivalent to each other are also elements of the same equivalence class.

28.6.2 Quotient set

Main article: Quotient set

The set of all possible equivalence classes of X by ~, denoted X/∼ := {[x] | x ∈ X} , is the quotient set of X by~. If X is a topological space, there is a natural way of transforming X/~ into a topological space; see quotient spacefor the details.

28.6.3 Projection

Main article: Projection (relational algebra)

The projection of ~ is the function π : X → X/∼ defined by π(x) = [x] which maps elements of X into theirrespective equivalence classes by ~.

Theorem on projections:[1] Let the function f: X → B be such that a ~ b → f(a) = f(b). Then there is aunique function g : X/~ → B, such that f = gπ. If f is a surjection and a ~ b ↔ f(a) = f(b), then g is abijection.

28.6.4 Equivalence kernel

The equivalence kernel of a function f is the equivalence relation ~ defined by x ∼ y ⇐⇒ f(x) = f(y) . Theequivalence kernel of an injection is the identity relation.

28.6.5 Partition

Main article: Partition of a set







https://en.wikipedia.org/wiki/Projection_(relational_algebra)

https://en.wikipedia.org/wiki/Projection_(set_theory)






28.7. FUNDAMENTAL THEOREM OF EQUIVALENCE RELATIONS 97

A partition of X is a set P of nonempty subsets of X, such that every element of X is an element of a single elementof P. Each element of P is a cell of the partition. Moreover, the elements of P are pairwise disjoint and their unionis X.

Counting possible partitions

Let X be a finite set with n elements. Since every equivalence relation over X corresponds to a partition of X, andvice versa, the number of possible equivalence relations on X equals the number of distinct partitions of X, which isthe nth Bell number Bn:

Bn =1

e

∞∑k=0

kn

k!,

where the above is one of the ways to write the nth Bell number.

28.7 Fundamental theorem of equivalence relations

A key result links equivalence relations and partitions:[2][3][4]

• An equivalence relation ~ on a set X partitions X.

• Conversely, corresponding to any partition of X, there exists an equivalence relation ~ on X.

In both cases, the cells of the partition of X are the equivalence classes of X by ~. Since each element of X belongsto a unique cell of any partition of X, and since each cell of the partition is identical to an equivalence class of X by~, each element of X belongs to a unique equivalence class of X by ~. Thus there is a natural bijection from the setof all possible equivalence relations on X and the set of all partitions of X.

28.8 Comparing equivalence relations

If ~ and ≈ are two equivalence relations on the same set S, and a~b implies a≈b for all a,b ∈ S, then ≈ is said to be acoarser relation than ~, and ~ is a finer relation than ≈. Equivalently,

• ~ is finer than ≈ if every equivalence class of ~ is a subset of an equivalence class of ≈, and thus every equivalenceclass of ≈ is a union of equivalence classes of ~.

• ~ is finer than ≈ if the partition created by ~ is a refinement of the partition created by ≈.

The equality equivalence relation is the finest equivalence relation on any set, while the trivial relation that makes allpairs of elements related is the coarsest.The relation "~ is finer than ≈" on the collection of all equivalence relations on a fixed set is itself a partial orderrelation.

28.9 Generating equivalence relations

• Given any set X, there is an equivalence relation over the set [X→X] of all possible functions X→X. Two suchfunctions are deemed equivalent when their respective sets of fixpoints have the same cardinality, correspondingto cycles of length one in a permutation. Functions equivalent in this manner form an equivalence class on[X→X], and these equivalence classes partition [X→X].

https://en.wikipedia.org/wiki/Pairwise_disjoint


https://en.wikipedia.org/wiki/Bell_numbers

https://en.wikipedia.org/wiki/Dobinski%2527s_formula



https://en.wikipedia.org/wiki/Fixpoint




• An equivalence relation ~ on X is the equivalence kernel of its surjective projection π : X → X/~.[5] Conversely,any surjection between sets determines a partition on its domain, the set of preimages of singletons in thecodomain. Thus an equivalence relation over X, a partition of X, and a projection whose domain is X, are threeequivalent ways of specifying the same thing.

• The intersection of any collection of equivalence relations over X (binary relations viewed as a subset of X × X)is also an equivalence relation. This yields a convenient way of generating an equivalence relation: given anybinary relation R on X, the equivalence relation generated by R is the smallest equivalence relation containingR. Concretely, R generates the equivalence relation a ~ b if and only if there exist elements x1, x2, ..., xn in Xsuch that a = x1, b = xn, and (xi,xi₊ ₁)∈R or (xi₊₁,xi)∈R, i = 1, ..., n−1.

Note that the equivalence relation generated in this manner can be trivial. For instance, the equivalencerelation ~ generated by:

• • Any total order on X has exactly one equivalence class, X itself, because x ~ y for all x and y;• Any subset of the identity relation on X has equivalence classes that are the singletons of X.

• Equivalence relations can construct new spaces by “gluing things together.” Let X be the unit Cartesian square[0,1] × [0,1], and let ~ be the equivalence relation on X defined by ∀a, b ∈ [0,1] ((a, 0) ~ (a, 1) ∧ (0, b) ~ (1, b)).Then the quotient space X/~ can be naturally identified (homeomorphism) with a torus: take a square piece ofpaper, bend and glue together the upper and lower edge to form a cylinder, then bend the resulting cylinder soas to glue together its two open ends, resulting in a torus.

28.10 Algebraic structure

Much of mathematics is grounded in the study of equivalences, and order relations. Lattice theory captures themathematical structure of order relations. Even though equivalence relations are as ubiquitous in mathematics asorder relations, the algebraic structure of equivalences is not as well known as that of orders. The former structuredraws primarily on group theory and, to a lesser extent, on the theory of lattices, categories, and groupoids.

28.10.1 Group theory

Just as order relations are grounded in ordered sets, sets closed under pairwise supremum and infimum, equivalencerelations are grounded in partitioned sets, which are sets closed under bijections and preserve partition structure.Since all such bijections map an equivalence class onto itself, such bijections are also known as permutations. Hencepermutation groups (also known as transformation groups) and the related notion of orbit shed light on the mathe-matical structure of equivalence relations.Let '~' denote an equivalence relation over some nonempty set A, called the universe or underlying set. Let G denotethe set of bijective functions over A that preserve the partition structure of A: ∀x ∈ A ∀g ∈ G (g(x) ∈ [x]). Then thefollowing three connected theorems hold:[6]

• ~ partitions A into equivalence classes. (This is the Fundamental Theorem of Equivalence Relations, mentionedabove);

• Given a partition of A, G is a transformation group under composition, whose orbits are the cells of the parti-tion‡;

• Given a transformation group G over A, there exists an equivalence relation ~ over A, whose equivalence classesare the orbits of G.[7][8]

In sum, given an equivalence relation ~ over A, there exists a transformation group G over A whose orbits are theequivalence classes of A under ~.This transformation group characterisation of equivalence relations differs fundamentally from the way lattices char-acterize order relations. The arguments of the lattice theory operations meet and join are elements of some universe

https://en.wikipedia.org/wiki/Equivalence_relation#Equivalence_kernel


https://en.wikipedia.org/wiki/Equivalence_relation#Projection




https://en.wikipedia.org/wiki/Singleton_(mathematics)






https://en.wikipedia.org/wiki/Cartesian_square



https://en.wikipedia.org/wiki/Torus





https://en.wikipedia.org/wiki/Groupoid




https://en.wikipedia.org/wiki/Infimum




https://en.wikipedia.org/wiki/Permutation_group


https://en.wikipedia.org/wiki/Orbit_(group_theory)

https://en.wikipedia.org/wiki/Universe_(mathematics)

https://en.wikipedia.org/wiki/Partitions_of_a_set

https://en.wikipedia.org/wiki/Transformation_group


https://en.wikipedia.org/wiki/Meet_(mathematics)

https://en.wikipedia.org/wiki/Join_(mathematics)

28.11. EQUIVALENCE RELATIONS AND MATHEMATICAL LOGIC 99

A. Meanwhile, the arguments of the transformation group operations composition and inverse are elements of a setof bijections, A → A.Moving to groups in general, let H be a subgroup of some group G. Let ~ be an equivalence relation on G, such that a~ b ↔ (ab−1 ∈ H). The equivalence classes of ~—also called the orbits of the action of H on G—are the right cosetsof H in G. Interchanging a and b yields the left cosets.‡Proof.[9] Let function composition interpret group multiplication, and function inverse interpret group inverse. ThenG is a group under composition, meaning that ∀x ∈ A ∀g ∈ G ([g(x)] = [x]), because G satisfies the following fourconditions:

• G is closed under composition. The composition of any two elements of G exists, because the domain andcodomain of any element of G is A. Moreover, the composition of bijections is bijective;[10]

• Existence of identity function. The identity function, I(x)=x, is an obvious element of G;• Existence of inverse function. Every bijective function g has an inverse g−1, such that gg−1 = I;• Composition associates. f(gh) = (fg)h. This holds for all functions over all domains.[11]

Let f and g be any two elements of G. By virtue of the definition of G, [g(f(x))] = [f(x)] and [f(x)] = [x], so that[g(f(x))] = [x]. Hence G is also a transformation group (and an automorphism group) because function compositionpreserves the partitioning of A. □Related thinking can be found in Rosen (2008: chpt. 10).

28.10.2 Categories and groupoids

Let G be a set and let "~" denote an equivalence relation over G. Then we can form a groupoid representing thisequivalence relation as follows. The objects are the elements of G, and for any two elements x and y of G, there existsa unique morphism from x to y if and only if x~y.The advantages of regarding an equivalence relation as a special case of a groupoid include:

• Whereas the notion of “free equivalence relation” does not exist, that of a free groupoid on a directed graphdoes. Thus it is meaningful to speak of a “presentation of an equivalence relation,” i.e., a presentation of thecorresponding groupoid;

• Bundles of groups, group actions, sets, and equivalence relations can be regarded as special cases of the notionof groupoid, a point of view that suggests a number of analogies;

• In many contexts “quotienting,” and hence the appropriate equivalence relations often called congruences, areimportant. This leads to the notion of an internal groupoid in a category.[12]

28.10.3 Lattices

The possible equivalence relations on any set X, when ordered by set inclusion, form a complete lattice, called ConX by convention. The canonical map ker: X^X → Con X, relates the monoid X^X of all functions on X and Con X.ker is surjective but not injective. Less formally, the equivalence relation ker on X, takes each function f: X→X toits kernel ker f. Likewise, ker(ker) is an equivalence relation on X^X.

28.11 Equivalence relations and mathematical logic

Equivalence relations are a ready source of examples or counterexamples. For example, an equivalence relation withexactly two infinite equivalence classes is an easy example of a theory which is ω-categorical, but not categorical forany larger cardinal number.An implication of model theory is that the properties defining a relation can be proved independent of each other(and hence necessary parts of the definition) if and only if, for each property, examples can be found of relationsnot satisfying the given property while satisfying all the other properties. Hence the three defining properties ofequivalence relations can be proved mutually independent by the following three examples:



https://en.wikipedia.org/wiki/Bijections

https://en.wikipedia.org/wiki/Subgroup







https://en.wikipedia.org/wiki/Bijective





https://en.wikipedia.org/wiki/Associativity

https://en.wikipedia.org/wiki/Automorphism_group

https://en.wikipedia.org/wiki/Groupoid


https://en.wikipedia.org/wiki/Free_object




https://en.wikipedia.org/wiki/Set_inclusion

https://en.wikipedia.org/wiki/Complete_lattice







https://en.wikipedia.org/wiki/Morley%2527s_categoricity_theorem


https://en.wikipedia.org/wiki/Model_theory


• Reflexive and transitive: The relation ≤ on N. Or any preorder;

• Symmetric and transitive: The relation R on N, defined as aRb ↔ ab ≠ 0. Or any partial equivalence relation;

• Reflexive and symmetric: The relation R on Z, defined as aRb ↔ "a − b is divisible by at least one of 2 or 3.”Or any dependency relation.

Properties definable in first-order logic that an equivalence relation may or may not possess include:

• The number of equivalence classes is finite or infinite;

• The number of equivalence classes equals the (finite) natural number n;

• All equivalence classes have infinite cardinality;

• The number of elements in each equivalence class is the natural number n.

28.12 Euclidean relations

Euclid's The Elements includes the following “Common Notion 1":

Things which equal the same thing also equal one another.

Nowadays, the property described by Common Notion 1 is called Euclidean (replacing “equal” by “are in relationwith”). By “relation” is meant a binary relation, in which aRb is generally distinct from bRa. An Euclidean relationthus comes in two forms:

(aRc ∧ bRc) → aRb (Left-Euclidean relation)(cRa ∧ cRb) → aRb (Right-Euclidean relation)

The following theorem connects Euclidean relations and equivalence relations:

Theorem If a relation is (left or right) Euclidean and reflexive, it is also symmetric and transitive.

Proof for a left-Euclidean relation

(aRc ∧ bRc) → aRb [a/c] = (aRa ∧ bRa) → aRb [reflexive; erase T∧] = bRa → aRb. Hence R is symmetric.

(aRc ∧ bRc) → aRb [symmetry] = (aRc ∧ cRb) → aRb. Hence R is transitive. □

with an analogous proof for a right-Euclidean relation. Hence an equivalence relation is a relation that is Euclideanand reflexive. The Elements mentions neither symmetry nor reflexivity, and Euclid probably would have deemed thereflexivity of equality too obvious to warrant explicit mention.

28.13 See also

• Partition of a set

• Equivalence class

• Up to

• Conjugacy class

• Topological conjugacy

















https://en.wikipedia.org/wiki/Up_to

https://en.wikipedia.org/wiki/Conjugacy_class

https://en.wikipedia.org/wiki/Topological_conjugacy

28.14. NOTES 101

28.14 Notes[1] Garrett Birkhoff and Saunders Mac Lane, 1999 (1967). Algebra, 3rd ed. p. 35, Th. 19. Chelsea.

[2] Wallace, D. A. R., 1998. Groups, Rings and Fields. p. 31, Th. 8. Springer-Verlag.

[3] Dummit, D. S., and Foote, R. M., 2004. Abstract Algebra, 3rd ed. p. 3, Prop. 2. John Wiley & Sons.

[4] Karel Hrbacek & Thomas Jech (1999) Introduction to Set Theory, 3rd edition, pages 29–32, Marcel Dekker

[5] Garrett Birkhoff and Saunders Mac Lane, 1999 (1967). Algebra, 3rd ed. p. 33, Th. 18. Chelsea.

[6] Rosen (2008), pp. 243-45. Less clear is §10.3 of Bas van Fraassen, 1989. Laws and Symmetry. Oxford Univ. Press.

[7] Wallace, D. A. R., 1998. Groups, Rings and Fields. Springer-Verlag: 202, Th. 6.

[8] Dummit, D. S., and Foote, R. M., 2004. Abstract Algebra, 3rd ed. John Wiley & Sons: 114, Prop. 2.

[9] Bas van Fraassen, 1989. Laws and Symmetry. Oxford Univ. Press: 246.



[12] Borceux, F. and Janelidze, G., 2001. Galois theories, Cambridge University Press, ISBN 0-521-80309-8

28.15 References• Brown, Ronald, 2006. Topology and Groupoids. Booksurge LLC. ISBN 1-4196-2722-8.

• Castellani, E., 2003, “Symmetry and equivalence” in Brading, Katherine, and E. Castellani, eds., Symmetriesin Physics: Philosophical Reflections. Cambridge Univ. Press: 422-433.

• Robert Dilworth and Crawley, Peter, 1973. Algebraic Theory of Lattices. Prentice Hall. Chpt. 12 discusseshow equivalence relations arise in lattice theory.

• Higgins, P.J., 1971. Categories and groupoids. Van Nostrand. Downloadable since 2005 as a TAC Reprint.

• John Randolph Lucas, 1973. A Treatise on Time and Space. London: Methuen. Section 31.

• Rosen, Joseph (2008) Symmetry Rules: How Science and Nature are Founded on Symmetry. Springer-Verlag.Mostly chpts. 9,10.

• Raymond Wilder (1965) Introduction to the Foundations of Mathematics 2nd edition, Chapter 2-8: Axiomsdefining equivalence, pp 48–50, John Wiley & Sons.

28.16 External links• Hazewinkel, Michiel, ed. (2001), “Equivalence relation”, Encyclopedia of Mathematics, Springer, ISBN 978-

1-55608-010-4

• Bogomolny, A., "Equivalence Relationship" cut-the-knot. Accessed 1 September 2009

• Equivalence relation at PlanetMath

• Binary matrices representing equivalence relations at OEIS.



https://en.wikipedia.org/wiki/Karel_Hrbacek

https://en.wikipedia.org/wiki/Thomas_Jech

https://en.wikipedia.org/wiki/Marcel_Dekker



https://en.wikipedia.org/wiki/Bas_van_Fraassen


http://www.bangor.ac.uk/r.brown/topgpds.html


https://en.wikipedia.org/wiki/Robert_Dilworth


http://www.emis.de/journals/TAC/reprints/articles/7/tr7abs.html

https://en.wikipedia.org/wiki/John_Lucas_(philosopher)

https://en.wikipedia.org/wiki/Raymond_Wilder

https://en.wikipedia.org/wiki/John_Wiley_&_Sons

http://www.encyclopediaofmath.org/index.php?title=p/e036030






http://www.cut-the-knot.org/blue/equi.shtml

https://en.wikipedia.org/wiki/Cut-the-knot

http://planetmath.org/equivalencerelation

http://oeis.org/A231428


Logical matrices of the 52 equivalence relations on a 5-element set (Colored fields, including those in light gray, stand for ones; whitefields for zeros.)



Chapter 29

Euclidean relation

In mathematics, Euclidean relations are a class of binary relations that satisfy a weakened form of transitivity thatformalizes Euclid's “Common Notion 1” in The Elements: things which equal the same thing also equal one another.

29.1 Definition

A binary relation R on a set X is Euclidean (sometimes called right Euclidean) if it satisfies the following: for everya, b, c in X, if a is related to b and c, then b is related to c.[1]

To write this in predicate logic:

∀a, b, c ∈ X (aR b ∧ aR c→ bR c).

Dually, a relation R on X is left Euclidean if for every a, b, c in X, if b is related to a and c is related to a, then b isrelated to c:

∀a, b, c ∈ X (bR a ∧ cR a→ bR c).

29.2 Relation to transitivity

The property of being Euclidean is different from transitivity: both the Euclidean property and transitivity infer arelation between b and c from relations between a and b and between a and c, but with different argument orderingsin the relations. However, if a relation is symmetric, then the argument orders do not matter; thus a symmetric relationwith any one of these three properties (transitive, right Euclidean, left Euclidean) must have all three.[1]

If a relation is Euclidean and reflexive, then it must also be symmetric and hence transitive (following the previousparagraph), and so it must be an equivalence relation. Consequently, equivalence relations are exactly the reflexiveEuclidean relations.[1]

29.3 References[1] Fagin, Ronald (2003), Reasoning About Knowledge, MIT Press, p. 60, ISBN 978-0-262-56200-3.

103








https://en.wikipedia.org/wiki/Predicate_logic





https://en.wikipedia.org/wiki/Ronald_Fagin

http://books.google.com/books?id=xHmlRamoszMC&pg=PA60



Chapter 30

Exceptional isomorphism

In mathematics, an exceptional isomorphism, also called an accidental isomorphism, is an isomorphism betweenmembers ai and bj of two families (usually infinite) of mathematical objects, that is not an example of a pattern ofsuch isomorphisms.[note 1] These coincidences are at times considered a matter of trivia,[1] but in other respects theycan give rise to other phenomena, notably exceptional objects.[1] In the below, coincidences are listed in all placesthey occur.

30.1 Groups

30.1.1 Finite simple groups

The exceptional isomorphisms between the series of finite simple groups mostly involve projective special lineargroups and alternating groups, and are:[1]

• L2(4) ∼= L2(5) ∼= A5, the smallest non-abelian simple group (order 60);

• L2(7) ∼= L3(2), the second-smallest non-abelian simple group (order 168) – PSL(2,7);

• L2(9) ∼= A6,

• L4(2) ∼= A8,

• PSU4(2) ∼= PSp4(3), between a projective special orthogonal group and a projective symplectic group.

30.1.2 Groups of Lie type

In addition to the aforementioned, there are some isomorphisms involving SL, PSL, GL, PGL, and the natural mapsbetween these. For example, the groups over F5 have a number of exceptional isomorphisms:

• PSL(2, 5) ∼= A5∼= I, the alternating group on five elements, or equivalently the icosahedral group;

• PGL(2, 5) ∼= S5, the symmetric group on five elements;

• SL(2, 5) ∼= 2 · A5∼= 2I, the double cover of the alternating group A5, or equivalently the binary icosahedral

group.

30.1.3 Alternating groups and symmetric groups

There are coincidences between alternating groups and small groups of Lie type:

• L2(4) ∼= L2(5) ∼= A5,

104



https://en.wikipedia.org/wiki/Exceptional_object

https://en.wikipedia.org/wiki/Finite_simple_group

https://en.wikipedia.org/wiki/Projective_special_linear_group

https://en.wikipedia.org/wiki/Projective_special_linear_group

https://en.wikipedia.org/wiki/Alternating_group

https://en.wikipedia.org/wiki/PSL(2,7)

https://en.wikipedia.org/wiki/Projective_special_orthogonal_group

https://en.wikipedia.org/wiki/Projective_symplectic_group

https://en.wikipedia.org/wiki/Icosahedral_group


https://en.wikipedia.org/wiki/Covering_groups_of_the_alternating_and_symmetric_groups

https://en.wikipedia.org/wiki/Binary_icosahedral_group


30.1. GROUPS 105

The compound of five tetrahedra expresses the exceptional isomorphism between the icosahedral group and the alternating group onfive letters.

• L2(9) ∼= Sp4(2)′ ∼= A6,

• Sp4(2) ∼= S6,

• L4(2) ∼= O6(+, 2)′ ∼= A8,

• O6(+, 2) ∼= S8.

These can all be explained in a systematic way by using linear algebra (and the action of Sn on affine n -space) todefine the isomorphism going from the right side to the left side. (The above isomorphisms for A8 and S8 are linkedvia the exceptional isomorphism SL4/µ2

∼= SO6 .) There are also some coincidences with symmetries of regularpolyhedra: the alternating group A5 agrees with the icosahedral group (itself an exceptional object), and the doublecover of the alternating group A5 is the binary icosahedral group.

30.1.4 Cyclic groups

Cyclic groups of small order especially arise in various ways, for instance:

• C2∼= {±1} ∼= O(1) ∼= Spin(1) ∼= Z∗ , the last being the group of units of the integers

https://en.wikipedia.org/wiki/Compound_of_five_tetrahedra

https://en.wikipedia.org/wiki/Regular_polyhedra

https://en.wikipedia.org/wiki/Regular_polyhedra

https://en.wikipedia.org/wiki/Icosahedral_group




106 CHAPTER 30. EXCEPTIONAL ISOMORPHISM

30.1.5 Spheres

The spheres S0, S1, and S3 admit group structures, which arise in various ways:

• S0 ∼= O(1) ,

• S1 ∼= SO(2) ∼= U(1) ∼= Spin(2) ,

• S3 ∼= Spin(3) ∼= SU(2) ∼= Sp(1) .

30.1.6 Coxeter groups

B2 C2≅ ≅

D3A3 ≅

≅

≅

E4A4 ≅

E5D5 ≅

≅

The exceptional isomorphisms of connected Dynkin diagrams.

https://en.wikipedia.org/wiki/Dynkin_diagram

30.2. LIE THEORY 107

There are some exceptional isomorphisms of Coxeter diagrams, yielding isomorphisms of the corresponding Coxetergroups and of polytopes realizing the symmetries. These are:

• A2 = I2(2) (2-simplex is regular 3-gon/triangle);

• BC2 = I2(4) (2-cube (square) = 2-cross-polytope (diamond) = regular 4-gon)

• A3 = D3 (3-simplex (tetrahedron) is 3-demihypercube (demicube), as per diagram)

• A1 = B1 = C1 (= D1?)

• D2 = A1 × A1

• A4 = E4

• D5 = E5

Closely related ones occur in Lie theory for Dynkin diagrams.

30.2 Lie theory

In low dimensions, there are isomorphisms among the classical Lie algebras and classical Lie groups called acciden-tal isomorphisms. For instance, there are isomorphisms between low-dimensional spin groups and certain classicalLie groups, due to low-dimensional isomorphisms between the root systems of the different families of simple Liealgebras, visible as isomorphisms of the corresponding Dynkin diagrams:

• Trivially, A0 = B0 = C0 = D0

• A1 = B1 = C1 , or sl2 ∼= so3 ∼= sp1

• B2 = C2, or so5 ∼= sp2

• D2 = A1 × A1, or so4 ∼= sl2 ⊕ sl2 ; note that these are disconnected, but part of the D-series

• A3 = D3 sl4 ∼= so6

• A4 = E4; the E-series usually starts at 6, but can be started at 4, yielding isomorphisms

• D5 = E5

Spin(1) = O(1)Spin(2) = U(1) = SO(2)Spin(3) = Sp(1) = SU(2)Spin(4) = Sp(1) × Sp(1)Spin(5) = Sp(2)Spin(6) = SU(4)

30.3 See also• Exceptional object

• Mathematical coincidence, for numerical coincidences

30.4 References[1] Because these series of objects are presented differently, they are not identical objects (do not have identical descriptions),

but turn out to describe the same object, hence one refers to this as an isomorphism, not an equality (identity).

https://en.wikipedia.org/wiki/Coxeter_diagram

https://en.wikipedia.org/wiki/Spin_group

https://en.wikipedia.org/wiki/Root_systems

https://en.wikipedia.org/wiki/Simple_Lie_algebra

https://en.wikipedia.org/wiki/Simple_Lie_algebra

https://en.wikipedia.org/wiki/Orthogonal_group

https://en.wikipedia.org/wiki/Unitary_group

https://en.wikipedia.org/wiki/Special_orthogonal_group

https://en.wikipedia.org/wiki/Symplectic_group

https://en.wikipedia.org/wiki/Special_unitary_group




https://en.wikipedia.org/wiki/Special_unitary_group

https://en.wikipedia.org/wiki/Exceptional_object

https://en.wikipedia.org/wiki/Mathematical_coincidence

108 CHAPTER 30. EXCEPTIONAL ISOMORPHISM

30.5 References[1] Wilson, Robert A. (2009), “Chapter 1: Introduction”, The finite simple groups, Graduate Texts in Mathematics 251 251,

Berlin, New York: Springer-Verlag, doi:10.1007/978-1-84800-988-2, ISBN 978-1-84800-987-5, Zbl 05622792, 2007preprint; Chapter doi:10.1007/978-1-84800-988-2_1.

https://en.wikipedia.org/wiki/Robert_Arnott_Wilson

http://www.maths.qmul.ac.uk/~raw/fsgs_files/intro.ps

https://en.wikipedia.org/wiki/Graduate_Texts_in_Mathematics

https://en.wikipedia.org/wiki/Springer-Verlag


https://dx.doi.org/10.1007%252F978-1-84800-988-2



https://en.wikipedia.org/wiki/Zentralblatt_MATH

https://zbmath.org/?format=complete&q=an:05622792

http://www.maths.qmul.ac.uk/~raw/fsgs.html

http://www.maths.qmul.ac.uk/~raw/fsgs.html


https://dx.doi.org/10.1007%252F978-1-84800-988-2_1

Chapter 31

Fiber (mathematics)

In mathematics, the term fiber (or fibre in British English) can have two meanings, depending on the context:

1. In naive set theory, the fiber of the element y in the set Y under a map f : X → Y is the inverse image of thesingleton {y} under f.

2. In algebraic geometry, the notion of a fiber of a morphism of schemes must be defined more carefully because,in general, not every point is closed.

31.1 Definitions

31.1.1 Fiber in naive set theory

Let f : X → Y be a map. The fiber of an element y ∈ Y , commonly denoted by f−1(y) , is defined as

f−1({y}) = {x ∈ X | f(x) = y}.

In various applications, this is also called:

• the inverse image of {y} under the map f

• the preimage of {y} under the map f

• the level set of the function f at the point y.

The term level set is only used if f maps into the real numbers and so y is simply a number. If f is a continuousfunction and if y is in the image of f, then the level set of y under f is a curve in 2D, a surface in 3D, and moregenerally a hypersurface of dimension d-1.

31.1.2 Fiber in algebraic geometry

In algebraic geometry, if f : X → Y is a morphism of schemes, the fiber of a point p in Y is the fibered productX ×Y Spec k(p) where k(p) is the residue field at p.

31.2 Terminological variance

The recommended practice is to use the terms fiber, inverse image, preimage, and level set as follows:

• the fiber of the element y under the map f

109



https://en.wikipedia.org/wiki/Naive_set_theory






https://en.wikipedia.org/wiki/Algebraic_geometry


https://en.wikipedia.org/wiki/Scheme_(mathematics)




https://en.wikipedia.org/wiki/Level_set

https://en.wikipedia.org/wiki/Continuous_function




https://en.wikipedia.org/wiki/Two-dimensional_space

https://en.wikipedia.org/wiki/Three-dimensional_space

https://en.wikipedia.org/wiki/Hypersurface

https://en.wikipedia.org/wiki/Dimension

https://en.wikipedia.org/wiki/Algebraic_geometry

110 CHAPTER 31. FIBER (MATHEMATICS)

• the inverse image of the set {y} under the map f

• the preimage of the set {y} under the map f

• the level set of the function f at the point y.

By abuse of language, the following terminology is sometimes used but should be avoided:

• the fiber of the map f at the element y• the inverse image of the map f at the element y• the preimage of the map f at the element y• the level set of the point y under the map f.

31.3 See also• Fibration

• Fiber bundle

• Fiber product

• Image (category theory)

• Image (mathematics)

• Inverse relation

• Kernel (mathematics)

• Level set

• Preimage

• Relation

• Zero set

https://en.wikipedia.org/wiki/Abuse_of_terminology

https://en.wikipedia.org/wiki/Fibration

https://en.wikipedia.org/wiki/Fiber_bundle

https://en.wikipedia.org/wiki/Fiber_product

https://en.wikipedia.org/wiki/Image_(category_theory)







https://en.wikipedia.org/wiki/Zero_set

Chapter 32

Finitary relation

This article is about the set-theoretic notion of relation. For the common case, see binary relation.For other uses, see Relation (disambiguation).

In mathematics, a finitary relation has a finite number of “places”. In set theory and logic, a relation is a propertythat assigns truth values to k -tuples of individuals. Typically, the property describes a possible connection betweenthe components of a k -tuple. For a given set of k -tuples, a truth value is assigned to each k -tuple according towhether the property does or does not hold.An example of a ternary relation (i.e., between three individuals) is: "X was introduced to Y byZ ", where (X,Y, Z)is a 3-tuple of persons; for example, "Beatrice Wood was introduced to Henri-Pierre Roché by Marcel Duchamp" istrue, while "Karl Marx was introduced to Friedrich Engels by Queen Victoria" is false.

32.1 Informal introduction

Relation is formally defined in the next section. In this section we introduce the concept of a relation with a familiareveryday example. Consider the relation involving three roles that people might play, expressed in a statement of theform "X thinks that Y likes Z ". The facts of a concrete situation could be organized in a table like the following:Each row of the table records a fact or makes an assertion of the form "X thinks that Y likes Z ". For instance, thefirst row says, in effect, “Alice thinks that Bob likes Denise”. The table represents a relation S over the set P of peopleunder discussion:

P = {Alice, Bob, Charles, Denise}.

The data of the table are equivalent to the following set of ordered triples:

S = {(Alice, Bob, Denise), (Charles, Alice, Bob), (Charles, Charles, Alice), (Denise, Denise, Denise)}.

By a slight abuse of notation, it is usual to write S(Alice, Bob, Denise) to say the same thing as the first row ofthe table. The relation S is a ternary relation, since there are three items involved in each row. The relation itselfis a mathematical object defined in terms of concepts from set theory (i.e., the relation is a subset of the Cartesianproduct on {Person X, Person Y, Person Z}), that carries all of the information from the table in one neat package.Mathematically, then, a relation is simply an “ordered set”.The table for relation S is an extremely simple example of a relational database. The theoretical aspects of databasesare the specialty of one branch of computer science, while their practical impacts have become all too familiar in oureveryday lives. Computer scientists, logicians, and mathematicians, however, tend to see different things when theylook at these concrete examples and samples of the more general concept of a relation.For one thing, databases are designed to deal with empirical data, and experience is always finite, whereas mathematicsat the very least concerns itself with potential infinity. This difference in perspective brings up a number of ideas thatmay be usefully introduced at this point, if by no means covered in depth.

111


https://en.wikipedia.org/wiki/Relation_(disambiguation)


https://en.wikipedia.org/wiki/Logic

https://en.wikipedia.org/wiki/Truth_values



https://en.wikipedia.org/wiki/Ternary_relation

https://en.wikipedia.org/wiki/Beatrice_Wood

https://en.wikipedia.org/wiki/Henri-Pierre_Roch%C3%A9

https://en.wikipedia.org/wiki/Marcel_Duchamp

https://en.wikipedia.org/wiki/Karl_Marx

https://en.wikipedia.org/wiki/Friedrich_Engels

https://en.wikipedia.org/wiki/Queen_Victoria

https://en.wikipedia.org/wiki/Mathematical_object




https://en.wikipedia.org/wiki/Relational_database


112 CHAPTER 32. FINITARY RELATION

32.2 Relations with a small number of “places”

The variable k giving the number of "places" in the relation, 3 for the above example, is a non-negative integer,called the relation’s arity, adicity, or dimension. A relation with k places is variously called a k -ary, a k -adic, ora k -dimensional relation. Relations with a finite number of places are called finite-place or finitary relations. Itis possible to generalize the concept to include infinitary relations between infinitudes of individuals, for exampleinfinite sequences; however, in this article only finitary relations are discussed, which will from now on simply becalled relations.Since there is only one 0-tuple, the so-called empty tuple ( ), there are only two zero-place relations: the one thatalways holds, and the one that never holds. They are sometimes useful for constructing the base case of an inductionargument. One-place relations are called unary relations. For instance, any set (such as the collection of Nobellaureates) can be viewed as a collection of individuals having some property (such as that of having been awardedthe Nobel prize). Two-place relations are called binary relations or, in the past, dyadic relations. Binary relations arevery common, given the ubiquity of relations such as:

• Equality and inequality, denoted by signs such as ' = ' and ' < ' in statements like ' 5 < 12 ';

• Being a divisor of, denoted by the sign ' | ' in statements like ' 13 | 143 ';

• Set membership, denoted by the sign ' ∈ ' in statements like ' 1 ∈ N '.

A k -ary relation is a straightforward generalization of a binary relation.

32.3 Formal definitionsWhen two objects, qualities, classes, or attributes, viewed together by the mind, are seen under some

connexion, that connexion is called a relation.—Augustus De Morgan[1]

The simpler of the two definitions of k-place relations encountered in mathematics is:Definition 1. A relation L over the sets X1, …, Xk is a subset of their Cartesian product, written L ⊆ X1 × … × Xk.Relations are classified according to the number of sets in the defining Cartesian product, in other words, accordingto the number of terms following L. Hence:

• Lu denotes a unary relation or property;• Luv or uLv denote a binary relation;• Luvw denotes a ternary relation;• Luvwx denotes a quaternary relation.

Relations with more than four terms are usually referred to as k-ary or n-ary, for example, “a 5-ary relation”. A k-aryrelation is simply a set of k-tuples.The second definition makes use of an idiom that is common in mathematics, stipulating that “such and such is ann-tuple” in order to ensure that such and such a mathematical object is determined by the specification of n componentmathematical objects. In the case of a relation L over k sets, there are k + 1 things to specify, namely, the k sets plusa subset of their Cartesian product. In the idiom, this is expressed by saying that L is a (k + 1)-tuple.Definition 2. A relation L over the sets X1, …, Xk is a (k + 1)-tuple L = (X1, …, Xk, G(L)), where G(L) is a subsetof the Cartesian product X1 × … × Xk. G(L) is called the graph of L.Elements of a relation are more briefly denoted by using boldface characters, for example, the constant element a =(a1, …, ak) or the variable element x = (x1, …, xk).A statement of the form "a is in the relation L " or "a satisfies L " is taken to mean that a is in L under the firstdefinition and that a is in G(L) under the second definition.The following considerations apply under either definition:

https://en.wikipedia.org/wiki/Non-negative


https://en.wikipedia.org/wiki/Arity

https://en.wikipedia.org/wiki/Dimension

https://en.wikipedia.org/wiki/Finitary

https://en.wikipedia.org/wiki/Infinite_sequence

https://en.wikipedia.org/wiki/Mathematical_induction

https://en.wikipedia.org/wiki/Nobel_laureates

https://en.wikipedia.org/wiki/Nobel_laureates

https://en.wikipedia.org/wiki/Nobel_prize





https://en.wikipedia.org/wiki/Divisor


https://en.wikipedia.org/wiki/Augustus_De_Morgan



https://en.wikipedia.org/wiki/Unary_relation





32.4. HISTORY 113

• The sets Xj for j = 1 to k are called the domains of the relation. Under the first definition, the relation does notuniquely determine a given sequence of domains.

• If all of the domains Xj are the same set X, then it is simpler to refer to L as a k-ary relation over X.

• If any of the domains Xj is empty, then the defining Cartesian product is empty, and the only relation over sucha sequence of domains is the empty relation L = ∅ . Hence it is commonly stipulated that all of the domainsbe nonempty.

As a rule, whatever definition best fits the application at hand will be chosen for that purpose, and anything that fallsunder it will be called a relation for the duration of that discussion. If it becomes necessary to distinguish the twodefinitions, an entity satisfying the second definition may be called an embedded or included relation.If L is a relation over the domains X1, …, Xk, it is conventional to consider a sequence of terms called variables, x1,…, xk, that are said to range over the respective domains.Let a Boolean domain B be a two-element set, say, B = {0, 1}, whose elements can be interpreted as logical values,typically 0 = false and 1 = true. The characteristic function of the relation L, written ƒL or χ(L), is the Boolean-valuedfunction ƒL : X1 × … × Xk → B, defined in such a way that ƒL( x ) = 1 just in case the k-tuple x is in the relation L.Such a function can also be called an indicator function, particularly in probability and statistics, to avoid confusionwith the notion of a characteristic function in probability theory.It is conventional in applied mathematics, computer science, and statistics to refer to a Boolean-valued function like ƒLas a k-place predicate. From the more abstract viewpoint of formal logic and model theory, the relation L constitutesa logical model or a relational structure that serves as one of many possible interpretations of some k-place predicatesymbol.Because relations arise in many scientific disciplines as well as in many branches of mathematics and logic, thereis considerable variation in terminology. This article treats a relation as the set-theoretic extension of a relationalconcept or term. A variant usage reserves the term “relation” to the corresponding logical entity, either the logicalcomprehension, which is the totality of intensions or abstract properties that all of the elements of the relation inextension have in common, or else the symbols that are taken to denote these elements and intensions. Further, somewriters of the latter persuasion introduce terms with more concrete connotations, like “relational structure”, for theset-theoretic extension of a given relational concept.

32.4 History

The logician Augustus De Morgan, in work published around 1860, was the first to articulate the notion of relationin anything like its present sense. He also stated the first formal results in the theory of relations (on De Morgan andrelations, see Merrill 1990). Charles Sanders Peirce restated and extended De Morgan’s results. Bertrand Russell(1938; 1st ed. 1903) was historically important, in that it brought together in one place many 19th century results onrelations, especially orders, by Peirce, Gottlob Frege, Georg Cantor, Richard Dedekind, and others. Russell and A.N. Whitehead made free use of these results in their Principia Mathematica.

32.5 Notes[1] De Morgan, A. (1858) “On the syllogism, part 3” in Heath, P., ed. (1966) On the syllogism and other logical writings.

Routledge. P. 119,

32.6 See also• Correspondence (mathematics)

• Functional relation


• Hypergraph

https://en.wikipedia.org/wiki/Domain_of_a_relation

https://en.wikipedia.org/wiki/Boolean_domain


https://en.wikipedia.org/wiki/Boolean-valued_function

https://en.wikipedia.org/wiki/Boolean-valued_function

https://en.wikipedia.org/wiki/Characteristic_function_(probability_theory)


https://en.wikipedia.org/wiki/Predicate_(mathematics)

https://en.wikipedia.org/wiki/Formal_logic


https://en.wikipedia.org/wiki/Interpretation_(logic)




https://en.wikipedia.org/wiki/Extension_(semantics)

https://en.wikipedia.org/wiki/Comprehension_(logic)

https://en.wikipedia.org/wiki/Comprehension_(logic)

https://en.wikipedia.org/wiki/Intension


https://en.wikipedia.org/wiki/Charles_Sanders_Peirce

https://en.wikipedia.org/wiki/Bertrand_Russell


https://en.wikipedia.org/wiki/Gottlob_Frege

https://en.wikipedia.org/wiki/Georg_Cantor

https://en.wikipedia.org/wiki/Richard_Dedekind

https://en.wikipedia.org/wiki/A._N._Whitehead

https://en.wikipedia.org/wiki/A._N._Whitehead


https://en.wikipedia.org/wiki/Correspondence_(mathematics)



https://en.wikipedia.org/wiki/Hypergraph

114 CHAPTER 32. FINITARY RELATION


• Logical matrix

• Partial order

• Projection (set theory)

• Reflexive relation

• Relation algebra

• Sign relation

• Transitive relation

• Relational algebra

• Relational model

32.7 References• Peirce, C.S. (1870), “Description of a Notation for the Logic of Relatives, Resulting from an Amplification

of the Conceptions of Boole’s Calculus of Logic”, Memoirs of the American Academy of Arts and Sciences 9,317–78, 1870. Reprinted, Collected Papers CP 3.45–149, Chronological Edition CE 2, 359–429.

• Ulam, S.M. and Bednarek, A.R. (1990), “On the Theory of Relational Structures and Schemata for ParallelComputation”, pp. 477–508 in A.R. Bednarek and Françoise Ulam (eds.), Analogies Between Analogies: TheMathematical Reports of S.M. Ulam andHis Los Alamos Collaborators, University of California Press, Berkeley,CA.

32.8 Bibliography• Bourbaki, N. (1994) Elements of the History of Mathematics, John Meldrum, trans. Springer-Verlag.

• Carnap, Rudolf (1958) Introduction to Symbolic Logic with Applications. Dover Publications.

• Halmos, P.R. (1960) Naive Set Theory. Princeton NJ: D. Van Nostrand Company.

• Lawvere, F.W., and R. Rosebrugh (2003) Sets for Mathematics, Cambridge Univ. Press.

• Lucas, J. R. (1999) Conceptual Roots of Mathematics. Routledge.

• Maddux, R.D. (2006) Relation Algebras, vol. 150 in 'Studies in Logic and the Foundations of Mathematics’.Elsevier Science.

• Merrill, Dan D. (1990) Augustus De Morgan and the logic of relations. Kluwer.

• Peirce, C.S. (1984) Writings of Charles S. Peirce: A Chronological Edition, Volume 2, 1867-1871. PeirceEdition Project, eds. Indiana University Press.

• Russell, Bertrand (1903/1938) The Principles of Mathematics, 2nd ed. Cambridge Univ. Press.

• Suppes, Patrick (1960/1972) Axiomatic Set Theory. Dover Publications.

• Tarski, A. (1956/1983) Logic, Semantics, Metamathematics, Papers from 1923 to 1938, J.H. Woodger, trans.1st edition, Oxford University Press. 2nd edition, J. Corcoran, ed. Indianapolis IN: Hackett Publishing.

• Ulam, S.M. (1990) Analogies Between Analogies: The Mathematical Reports of S.M. Ulam and His Los AlamosCollaborators in A.R. Bednarek and Françoise Ulam, eds., University of California Press.

• R. Fraïssé, Theory of Relations (North Holland; 2000).







https://en.wikipedia.org/wiki/Sign_relation



https://en.wikipedia.org/wiki/Relational_model


https://en.wikipedia.org/wiki/Stanislaw_Ulam

https://en.wikipedia.org/wiki/Al_Bednarek

https://en.wikipedia.org/wiki/Nicolas_Bourbaki

https://en.wikipedia.org/wiki/Rudolf_Carnap

https://en.wikipedia.org/wiki/Paul_Richard_Halmos

https://en.wikipedia.org/wiki/Francis_William_Lawvere

https://en.wikipedia.org/wiki/John_Lucas_(philosopher)

https://en.wikipedia.org/wiki/Roger_Maddux



http://fair-use.org/bertrand-russell/the-principles-of-mathematics



https://en.wikipedia.org/wiki/Stanislaw_Ulam

Chapter 33

Foundational relation

In set theory, a foundational relation on a set or proper class lets each nonempty subset admit a relational minimalelement.Formally, let (A, R) be a binary relation structure, where A is a class (set or proper class), and R is a binary relationdefined on A. Then (A, R) is a foundational relation if and only if any nonempty subset in A has a R-minimal element.In predicate logic,

(∀S)(S ⊆ A ∧ S ̸= ∅ ⇒ (∃x ∈ S)(S ∩R−1{x} = ∅)

), [1]

in which ∅ denotes the empty set, and R−1{x} denotes the class of the elements that precede x in the relation R. Thatis,

R−1{x} = {y|yRx}. [2]

Here x is an R-minimal element in the subset S, since none of its R-predecessors is in S.

33.1 See also• Binary relation

• Well-order

33.2 References[1] See Definition 6.21 in Zaring W.M., G. Takeuti (1971). Introduction to axiomatic set theory (2nd, rev. ed.). New York:

Springer-Verlag. ISBN 0387900241.

[2] See Theorem 6.19 and Definition 6.20 in Zaring W.M., G. Takeuti (1971). Introduction to axiomatic set theory (2nd, rev.ed.). New York: Springer-Verlag. ISBN 0387900241.

115





https://en.wikipedia.org/wiki/Maximal_element



https://en.wikipedia.org/wiki/Predicate_logic








Chapter 34






116









































34.1.2 Example































34.3. RELATIONS OVER A SET 119





34.2.1 Difunctional



















































































34.4.1 Complement








34.4.2 Restriction




































































34.7. EXAMPLES OF COMMON BINARY RELATIONS 123









34.8 See also


• Hasse diagram



• Order theory


















































































https://dx.doi.org/10.1007%252F978-3-540-74788-8_18




https://dx.doi.org/10.1007%252F978-3-540-27778-1_15







https://dx.doi.org/10.1007%252F978-3-662-44124-4_7
























55608-010-4







Chapter 35

Hypostatic abstraction

Hypostatic abstraction in mathematical logic, also known as hypostasis or subjectal abstraction, is a formal op-eration that transforms a predicate into a relation; for example “Honey is sweet” is transformed into “Honey possessessweetness”. The relation is created between the original subject and a new term that represents the property expressedby the original predicate.Hypostasis changes a propositional formula of the form X is Y to another one of the form X has the property of beingY or X has Y-ness. The logical functioning of the second object Y-ness consists solely in the truth-values of thosepropositions that have the corresponding concrete term Y as the predicate. The object of thought introduced in thisway may be called a hypostatic object and in some senses an abstract object and a formal object.The above definition is adapted from the one given by Charles Sanders Peirce (CP 4.235, “The Simplest Mathematics”(1902), in Collected Papers, CP 4.227–323). As Peirce describes it, the main point about the formal operation ofhypostatic abstraction, insofar as it operates on formal linguistic expressions, is that it converts an adjective or predicateinto an extra subject, thus increasing by one the number of “subject” slots -- called the arity or adicity -- of the mainpredicate.The transformation of “honey is sweet” into “honey possesses sweetness” can be viewed in several ways:

The grammatical trace of this hypostatic transformation is a process that extracts the adjective “sweet” from thepredicate “is sweet”, replacing it by a new, increased-arity predicate “possesses”, and as a by-product of the reaction,as it were, precipitating out the substantive “sweetness” as a second subject of the new predicate.The abstraction of hypostasis takes the concrete physical sense of “taste” found in “honey is sweet” and gives it formalmetaphysical characteristics in “honey has sweetness”.

126


https://en.wikipedia.org/wiki/Formal_calculation

https://en.wikipedia.org/wiki/Formal_calculation


https://en.wikipedia.org/wiki/Relation_(philosophy)


https://en.wikipedia.org/wiki/Propositional_formula

https://en.wikipedia.org/wiki/Abstract_object


https://en.wikipedia.org/wiki/Arity

https://en.wikipedia.org/wiki/Metaphysical

35.1. SEE ALSO 127

35.1 See also

35.2 References• Peirce, C.S., Collected Papers of Charles Sanders Peirce, vols. 1–6 (1931–1935), Charles Hartshorne and Paul

Weiss, eds., vols. 7–8 (1958), Arthur W. Burks, ed., Harvard University Press, Cambridge, MA.

35.3 External links• J. Jay Zeman, Peirce on Abstraction


https://en.wikipedia.org/wiki/Charles_Sanders_Peirce_bibliography#CP

https://en.wikipedia.org/wiki/Charles_Hartshorne

https://en.wikipedia.org/wiki/Paul_Weiss_(philosopher)

https://en.wikipedia.org/wiki/Paul_Weiss_(philosopher)

https://en.wikipedia.org/wiki/Arthur_Burks

http://web.clas.ufl.edu/users/jzeman/peirce_on_abstraction.htm

Chapter 36

Idempotence

For the concept in matrix algebra, see Idempotent matrix.

Idempotence (/ˌaɪdɨmˈpoʊtəns/EYE-dəm-POH-təns) is the property of certain operations in mathematics and computerscience, that can be applied multiple times without changing the result beyond the initial application. The conceptof idempotence arises in a number of places in abstract algebra (in particular, in the theory of projectors and closureoperators) and functional programming (in which it is connected to the property of referential transparency).The term was introduced by Benjamin Peirce[1] in the context of elements of algebras that remain invariant whenraised to a positive integer power, and literally means "(the quality of having) the same power”, from idem + potence(same + power).There are several meanings of idempotence, depending on what the concept is applied to:

• A unary operation (or function) is idempotent if, whenever it is applied twice to any value, it gives the sameresult as if it were applied once; i.e., ƒ(ƒ(x)) ≡ ƒ(x). For example, the absolute value function, where abs(abs(x))≡ abs(x).

• A binary operation is idempotent if, whenever it is applied to two equal values, it gives that value as the result.For example, the function giving the maximum value of two equal values is idempotent: max (x, x) ≡ x.

• Given a binary operation, an idempotent element (or simply an “idempotent”) for the operation is a value forwhich the operation, when given that value for both of its operands, gives that value as the result. For example,the number 1 is an idempotent of multiplication: 1 × 1 = 1.

36.1 Definitions

36.1.1 Unary operation

A unary operation f , that is, a map from some set S into itself, is called idempotent if, for all x in S ,

f(f(x)) = f(x)

In particular, the identity function idS , defined by idS (x) = x , is idempotent, as is the constant functionKc , wherec is an element of S , defined by Kc (x) = c .An important class of idempotent functions is given by projections in a vector space. An example of a projection isthe function πxy defined by πxy (x, y, z) = (x, y, 0) , which projects an arbitrary point in 3D space to a point on thexy -plane, where the third coordinate ( z ) is equal to 0.A unary operation f : S → S is idempotent if it maps each element of S to a fixed point of f . We can partition aset with n elements into k chosen fixed points and n− k non-fixed points, and then kn−k is the number of differentidempotent functions. Hence, taking into account all possible partitions,

128

https://en.wikipedia.org/wiki/Idempotent_matrix

https://en.wikipedia.org/wiki/Help:IPA_for_English

https://en.wikipedia.org/wiki/Wikipedia:Pronunciation_respelling_key






https://en.wikipedia.org/wiki/Projector_(linear_algebra)

https://en.wikipedia.org/wiki/Closure_operator


https://en.wikipedia.org/wiki/Functional_programming

https://en.wikipedia.org/wiki/Referential_transparency_(computer_science)

https://en.wikipedia.org/wiki/Benjamin_Peirce

https://en.wiktionary.org/wiki/idem

https://en.wiktionary.org/wiki/potence

https://en.wikipedia.org/wiki/Unary_operation


https://en.wikipedia.org/wiki/Absolute_value


https://en.wikipedia.org/wiki/Maximum




https://en.wikipedia.org/wiki/Constant_function

https://en.wikipedia.org/wiki/Projection_(linear_algebra)


https://en.wikipedia.org/wiki/Plane_(mathematics)

https://en.wikipedia.org/wiki/Fixed_point_(mathematics)

36.2. COMMON EXAMPLES 129

n∑k=0

(n

k

)kn−k

is the total number of possible idempotent functions on the set. The integer sequence of the number of idempotentfunctions as given by the sum above for n = {0, 1, 2, . . . } starts with 1, 1, 3, 10, 41, 196, 1057, 6322, 41393, . . . .(sequence A000248 in OEIS)Neither the property of being idempotent nor that of being not is preserved under composition of unary functions.[2]

As an example for the former, f(x) = x mod 3 and g(x) = max(x,5) are both idempotent, but f∘g is not,[3] althoughg∘f happens to be.[4] As an example for the latter, the negation function ¬ on truth values isn't idempotent, but ¬∘¬is.

36.1.2 Idempotent elements and binary operations

Main article: Idempotent element

Given a binary operation ⋆ on a set S , an element x is said to be idempotent (with respect to ⋆ ) if:

x⋆x = x

In particular an identity element of ⋆ , if it exists, is idempotent with respect to the operation ⋆ . The binaryoperation itself is called idempotent if every element of S is idempotent. That is, for all x ∈ S when ∈ denotes setmembership:

x⋆x = x

For example, the operations of set union and set intersection are both idempotent, as are logical conjunction andlogical disjunction, and, in general, the meet and join operations of a lattice.

36.1.3 Connections

The connections between the three notions are as follows.

• The statement that the binary operation ★ on a set S is idempotent, is equivalent to the statement that everyelement of S is idempotent for ★.

• The defining property of unary idempotence, f(f(x)) = f(x) for x in the domain of f, can equivalently berewritten as f ∘ f = f, using the binary operation of function composition denoted by ∘. Thus, the statementthat f is an idempotent unary operation on S is equivalent to the statement that f is an idempotent element withrespect to the function composition operation ∘ on functions from S to S.

36.2 Common examples

36.2.1 Functions

As mentioned above, the identity map and the constant maps are always idempotent maps. The absolute value functionof a real or complex argument, and the floor function of a real argument are idempotent. The function that assignsto every subset U of some topological space X the closure of U is idempotent on the power set P (X) of X .It is an example of a closure operator; all closure operators are idempotent functions. The operation of subtractingthe average of a list of numbers from every number in the list is idempotent. For example, consider the numbers3, 6, 8, 8, and10 . The average

∑n1 xn

n ∀xn is 3+6+8+8+105 = 35

5 = 7 . Subtracting 7 from every number in the listyields (−4) , (−1) , 1, 1, 3 . The average

∑n1 xn

n ∀xn of that list is (−4)+(−1)+1+1+35 = 0

5 = 0 . Subtracting 0 fromevery number in that list yields the same list.

https://en.wikipedia.org/wiki/Integer_sequence



https://en.wikipedia.org/wiki/Modulo_arithmetic

https://en.wikipedia.org/wiki/Truth_value

https://en.wikipedia.org/wiki/Idempotent_element






https://en.wikipedia.org/wiki/Logical_disjunction

https://en.wikipedia.org/wiki/Meet_(mathematics)

https://en.wikipedia.org/wiki/Join_(mathematics)



https://en.wikipedia.org/wiki/Absolute_value



https://en.wikipedia.org/wiki/Floor_function


https://en.wikipedia.org/wiki/Closure_(topology)



130 CHAPTER 36. IDEMPOTENCE

36.2.2 Formal languages

The Kleene star and Kleene plus operators used to express repetition in formal languages are idempotent.

36.2.3 Idempotent ring elements

Main article: Idempotent element

An idempotent element of a ring is, by definition, an element that is idempotent for the ring’s multiplication.[5] Thatis, an element a is idempotent precisely when a2 = a.Idempotent elements of rings yield direct decompositions of modules, and play a role in describing other homologicalproperties of the ring. While “idempotent” usually refers to the multiplication operation of a ring, there are rings inwhich both operations are idempotent: Boolean algebras are such an example.

36.2.4 Other examples

In Boolean algebra, both the logical and and the logical or operations are idempotent. This implies that every elementof Boolean algebra is idempotent with respect to both of these operations. Specifically, x ∧ x = x and x ∨ x = xfor all x . In linear algebra, projections are idempotent. In fact, the projections of a vector space are exactly theidempotent elements of the ring of linear transformations of the vector space. After fixing a basis, it can be shownthat the matrix of a projection with respect to this basis is an idempotent matrix. An idempotent semiring (alsosometimes called a dioid) is a semiring whose addition (not multiplication) is idempotent. If both operations of thesemiring are idempotent, then the semiring is called doubly idempotent.[6]

36.3 Computer science meaning

See also: Referential transparency (computer science), Reentrant (subroutine) and Stable sort

In computer science, the term idempotent is used more comprehensively to describe an operation that will producethe same results if executed once or multiple times.[7] This may have a different meaning depending on the contextin which it is applied. In the case of methods or subroutine calls with side effects, for instance, it means that themodified state remains the same after the first call. In functional programming, though, an idempotent function isone that has the property f(f(x)) = f(x) for any value x.[8]

This is a very useful property in many situations, as it means that an operation can be repeated or retried as oftenas necessary without causing unintended effects. With non-idempotent operations, the algorithm may have to keeptrack of whether the operation was already performed or not.

36.3.1 Examples

Looking up some customer’s name and address in a database are typically idempotent (in fact nullipotent), since thiswill not cause the database to change. Similarly, changing a customer’s address is typically idempotent, because thefinal address will be the same no matter how many times it is submitted. However, placing an order for a car for thecustomer is typically not idempotent, since running the method/call several times will lead to several orders beingplaced. Canceling an order is idempotent, because the order remains canceled no matter how many requests aremade.A composition of idempotent methods or subroutines, however, is not necessarily idempotent if a later method inthe sequence changes a value that an earlier method depends on – idempotence is not closed under composition. Forexample, suppose the initial value of a variable is 3 and there is a sequence that reads the variable, then changes it to 5,and then reads it again. Each step in the sequence is idempotent: both steps reading the variable have no side effectsand changing a variable to 5 will always have the same effect no matter how many times it is executed. Nonetheless,executing the entire sequence once produces the output (3, 5), but executing it a second time produces the output (5,5), so the sequence is not idempotent.[9]

https://en.wikipedia.org/wiki/Kleene_star

https://en.wikipedia.org/wiki/Kleene_plus

https://en.wikipedia.org/wiki/Formal_grammar

https://en.wikipedia.org/wiki/Idempotent_element


https://en.wikipedia.org/wiki/Indecomposable_module

https://en.wikipedia.org/wiki/Module_(algebra)

https://en.wikipedia.org/wiki/Boolean_algebra

https://en.wikipedia.org/wiki/Boolean_algebra




https://en.wikipedia.org/wiki/Projection_(linear_algebra)

https://en.wikipedia.org/wiki/Linear_transformation

https://en.wikipedia.org/wiki/Basis_(linear_algebra)

https://en.wikipedia.org/wiki/Idempotent_semiring


https://en.wikipedia.org/wiki/Reentrant_(subroutine)

https://en.wikipedia.org/wiki/Stable_sort


https://en.wikipedia.org/wiki/Method_(computer_science)

https://en.wikipedia.org/wiki/Subroutine

https://en.wikipedia.org/wiki/Side_effect_(computer_science)

https://en.wikipedia.org/wiki/Functional_programming

https://en.wikipedia.org/wiki/Database

https://en.wiktionary.org/wiki/nullipotent

36.4. APPLIED EXAMPLES 131

In the HyperText Transfer Protocol (HTTP), idempotence and safety are the major attributes that separate HTTPverbs. Of the major HTTP verbs, GET, PUT, and DELETE are idempotent (if implemented according to the stan-dard), but POST is not.[9] These verbs represent very abstract operations in computer science: GET retrieves aresource; PUT stores content at a resource; and DELETE eliminates a resource. As in the example above, readingdata usually has no side effects, so it is idempotent (in fact nullipotent). Storing a given set of content is usuallyidempotent, as the final value stored remains the same after each execution. And deleting something is generallyidempotent, as the end result is always the absence of the thing deleted.In Event Stream Processing, idempotence refers to the ability of a system to produce the same outcome, even if anevent or message is received more than once.In a load-store architecture, instructions that might possibly cause a page fault are idempotent. So if a page faultoccurs, the OS can load the page from disk and then simply re-execute the faulted instruction. In a processor wheresuch instructions are not idempotent, dealing with page faults is much more complex.

36.4 Applied examples

Applied examples that many people could encounter in their day-to-day lives include elevator call buttons and cross-walk buttons.[10] The initial activation of the button moves the system into a requesting state, until the request issatisfied. Subsequent activations of the button between the initial activation and the request being satisfied have noeffect.

36.5 See also

• Closure operator

• Fixed point (mathematics)

• Idempotent of a code

• Nilpotent

• Idempotent matrix

• Idempotent relation — a generalization of idempotence to binary relations

• List of matrices

• Pure function

• Referential transparency (computer science)

• Iterated function

• Biordered set

• Involution (mathematics)

36.6 References[1] Polcino & Sehgal (2002), p. 127.

[2] If f and g commute, i.e. if f∘g = g∘f, then idempotency of both f and g implies that of f∘g, since f∘g ∘ f∘g = f∘f ∘ g∘g =f ∘ g, using the associativity of composition.

[3] e.g. f(g(7)) = f(7) = 1, but f(g(1)) = f(5) = 2 ≠ 1

[4] also showing that commutation of f and g is not a necessary condition for idempotency preservation

[5] See Hazewinkel et al. (2004), p. 2.

https://en.wikipedia.org/wiki/Hypertext_Transfer_Protocol

https://en.wikipedia.org/wiki/Hypertext_Transfer_Protocol#Safe_methods

https://en.wikipedia.org/wiki/Hypertext_Transfer_Protocol#Request_methods

https://en.wikipedia.org/wiki/Hypertext_Transfer_Protocol#Request_methods

https://en.wikipedia.org/wiki/Event_Stream_Processing

https://en.wikipedia.org/wiki/Load-store_architecture

https://en.wikipedia.org/wiki/Page_fault

https://en.wikipedia.org/wiki/Motorola_68000#Interrupts

https://en.wikipedia.org/wiki/Elevator


https://en.wikipedia.org/wiki/Fixed_point_(mathematics)

https://en.wikipedia.org/wiki/Idempotent_of_a_code

https://en.wikipedia.org/wiki/Nilpotent

https://en.wikipedia.org/wiki/Idempotent_matrix

https://en.wikipedia.org/wiki/Idempotent_relation

https://en.wikipedia.org/wiki/List_of_matrices

https://en.wikipedia.org/wiki/Pure_function


https://en.wikipedia.org/wiki/Iterated_function

https://en.wikipedia.org/wiki/Biordered_set


https://en.wikipedia.org/wiki/Necessary_condition

https://en.wikipedia.org/wiki/Michiel_Hazewinkel

132 CHAPTER 36. IDEMPOTENCE

[6] Gondran & Minoux. Graphs, dioids and semirings. Springer, 2008, p. 34

[7] Rodriguez, Alex. “RESTful Web services: The basics”. IBM developerWorks. IBM. Retrieved 24 April 2013.

[8] http://foldoc.org/idempotent

[9] IETF, Hypertext Transfer Protocol (HTTP/1.1): Semantics and Content. See also HyperText Transfer Protocol.

[10] http://web.archive.org/web/20110523081716/http://www.nclabor.com/elevator/geartrac.pdf For example, this design spec-ification includes detailed algorithm for when elevator cars will respond to subsequent calls for service

36.7 Further reading• “idempotent” at FOLDOC

• Goodearl, K. R. (1991), von Neumann regular rings (2 ed.), Malabar, FL: Robert E. Krieger Publishing Co.Inc., pp. xviii+412, ISBN 0-89464-632-X, MR 1150975 (93m:16006)

• Gunawardena, Jeremy (1998), “An introduction to idempotency”, in Gunawardena, Jeremy, Idempotency.Based on a workshop, Bristol, UK, October 3–7, 1994 (PDF), Cambridge: Cambridge University Press, pp.1–49, Zbl 0898.16032

• Hazewinkel, Michiel; Gubareni, Nadiya; Kirichenko, V. V. (2004), Algebras, rings and modules. vol. 1,Mathematics and its Applications 575, Dordrecht: Kluwer Academic Publishers, pp. xii+380, ISBN 1-4020-2690-0, MR 2106764 (2006a:16001)

• Lam, T. Y. (2001), A first course in noncommutative rings, Graduate Texts in Mathematics 131 (2 ed.), NewYork: Springer-Verlag, pp. xx+385, ISBN 0-387-95183-0, MR 1838439 (2002c:16001)

• Lang, Serge (1993), Algebra (Third ed.), Reading, Mass.: Addison-Wesley Pub. Co., ISBN 978-0-201-55540-0, Zbl 0848.13001 p. 443

• Peirce, Benjamin. Linear Associative Algebra 1870.

• Polcino Milies, César; Sehgal, Sudarshan K. (2002), An introduction to group rings, Algebras and Applications1, Dordrecht: Kluwer Academic Publishers, pp. xii+371, ISBN 1-4020-0238-6, MR 1896125 (2003b:16026)

36.8 External links• Hazewinkel, Michiel, ed. (2001), “Idempotent”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-

010-4

https://www.ibm.com/developerworks/webservices/library/ws-restful/

http://foldoc.org/idempotent

http://tools.ietf.org/html/rfc7231#section-4.2.2

https://en.wikipedia.org/wiki/Hypertext_Transfer_Protocol

http://web.archive.org/web/20110523081716/http://www.nclabor.com/elevator/geartrac.pdf

http://foldoc.org/idempotent

https://en.wikipedia.org/wiki/FOLDOC




https://www.ams.org/mathscinet-getitem?mr=1150975+%252893m%253A16006%2529

http://www.hpl.hp.com/techreports/96/HPL-BRIMS-96-24.pdf

http://www.hpl.hp.com/techreports/96/HPL-BRIMS-96-24.pdf

https://en.wikipedia.org/wiki/Cambridge_University_Press


https://zbmath.org/?format=complete&q=an:0898.16032

https://en.wikipedia.org/wiki/Hazewinkel,_Michiel





https://www.ams.org/mathscinet-getitem?mr=2106764+%25282006a%253A16001%2529




https://www.ams.org/mathscinet-getitem?mr=1838439+%25282002c%253A16001%2529

https://en.wikipedia.org/wiki/Serge_Lang

https://en.wikipedia.org/wiki/Addison-Wesley






http://www.math.harvard.edu/history/peirce_algebra/index.html




https://www.ams.org/mathscinet-getitem?mr=1896125+%25282003b%253A16026%2529

http://www.encyclopediaofmath.org/index.php?title=p/i050080






Chapter 37

Idempotent relation

In mathematics, an idempotent binary relation R ⊆ X × X is one for which R ∘ R = R.[1][2] This notion generalizesthat of an idempotent function to relations. Each idempotent relation is necessarily transitive, as the latter means R ∘R ⊆ R.For example, the relation < on ℚ is idempotent. In contrast, < on ℤ is not, since (<) ∘ (<) ⊇ (<) does not hold: e.g. 1< 2, but 1 < x < 2 is false for every x ∈ ℤ.Idempotent relations have been used as an example to illustrate the application of Mechanized Formalisation of math-ematics using the interactive theorem prover Isabelle/HOL. Besides checking the mathematical properties of finiteidempotent relations, an algorithm for counting the number of idempotent relations has been derived in Isabelle/HOL.[3][4]

37.1 References[1] Florian Kammüller, J. W. Sanders (2004). Idempotent Relation in Isabelle/HOL (PDF) (Technical report). TU Berlin. p.

27. 2004-04. Here:p.3

[2] Florian Kammüller (2011). “Mechanical Analysis of Finite Idempotent Relations”. Fundamenta Informaticae 107. pp.43–65. doi:10.3233/FI-2011-392.

[3] Florian Kammüller (2006). “Number of idempotent relations on n labeled elements”. The On-Line Ecyclopedea of IntegerSequences (A12137).

[4] Florian Kammüller (2008). Counting Idempotent Relations (PDF) (Technical report). TU Berlin. p. 27. 2008-15.

• Berstel, Jean; Perrin, Dominique; Reutenauer, Christophe (2010). Codes and automata. Encyclopedia ofMathematics and its Applications 129. Cambridge: Cambridge University Press. p. 330. ISBN 978-0-521-88831-8. Zbl 1187.94001.

133



https://en.wikipedia.org/wiki/Relation_composition

https://en.wikipedia.org/wiki/Idempotent_function



https://en.wikipedia.org/wiki/Integer_number

https://en.wikipedia.org/wiki/Superset

http://cs.tu-berlin.de/cs/ifb/Ahmed/RoteReihe/2004/TR2004_04.pdf


https://dx.doi.org/10.3233%252FFI-2011-392

https://oeis.org/search?q=11%252C123%252C2360&sort=&language=english&go=Search

http://www.eecs.tu-berlin.de/fileadmin/f4/TechReports/2008/2008-15.pdf

http://www-igm.univ-mlv.fr/~berstel/LivreCodes/Codes.html







Chapter 38

Intransitivity

This article is about intransitivity in mathematics. For the linguistics sense, see Intransitive verb.

In mathematics, intransitivity (sometimes called nontransitivity) is a property of binary relations that are nottransitive relations. This may include any relation that is not transitive, or the stronger property of antitransitiv-ity, which describes a relation that is never transitive.

38.1 Intransitivity

A relation is transitive if, whenever it relates some A to some B, and that B to some C, it also relates that A to that C.Some authors call a relation intransitive if it is not transitive, i.e. (if the relation in question is named R )

¬ (∀a, b, c : aRb ∧ bRc⇒ aRc) .

This statement is equivalent to

∃a, b, c : aRb ∧ bRc ∧ ¬(aRc)

For instance, in the food chain, wolves feed on deer, and deer feed on grass, but wolves do not feed on grass.[1] Thus,the feed on relation among life forms is intransitive, in this sense.Another example that does not involve preference loops arises in freemasonry: it may be the case that lodge Arecognizes lodge B, and lodge B recognizes lodge C, but lodge A does not recognize lodge C. Thus the recognitionrelation among Masonic lodges is intransitive.

38.2 Antitransitivity

Often the term intransitive is used to refer to the stronger property of antitransitivity.We just saw that the feed on relation is not transitive, but it still contains some transitivity: for instance: humans feedon rabbits, rabbits feed on carrots, and humans also feed on carrots.A relation is antitransitive if this never occurs at all, i.e.,

∀a, b, c : aRb ∧ bRc⇒ ¬aRc

Many authors use the term intransitivity to mean antitransitivity.[2][3]

An example of an antitransitive relation: the defeated relation in knockout tournaments. If player A defeated playerB and player B defeated player C, A can have never played C, and therefore, A has not defeated C.

134

https://en.wikipedia.org/wiki/Intransitive_verb




https://en.wikipedia.org/wiki/Mathematical_jargon#stronger

https://en.wikipedia.org/wiki/Food_chain

https://en.wikipedia.org/wiki/Freemasonry

https://en.wikipedia.org/wiki/Mathematical_jargon#stronger

https://en.wikipedia.org/wiki/Knockout_tournament

38.3. CYCLES 135

38.3 Cycles

The term intransitivity is often used when speaking of scenarios in which a relation describes the relative preferencesbetween pairs of options, and weighing several options produces a “loop” of preference:

• A is preferred to B

• B is preferred to C

• C is preferred to A

Rock, paper, scissors; Nontransitive dice; and Penney’s game are examples.Assuming no option is preferred to itself i.e. the relation is irreflexive, a preference relation with a loop is not transitive.For if it is, each option in the loop is preferred to each option, including itself. This can be illustrated for this exampleof a loop among A, B, and C. Assume the relation is transitive. Then, since A is preferred to B and B is preferred toC, also A is preferred to C. But then, since C is preferred to A, also A is preferred to A.Therefore such a preference loop (or "cycle") is known as an intransitivity.Notice that a cycle is neither necessary nor sufficient for a binary relation to be not transitive. For example, anequivalence relation possesses cycles but is transitive. Now, consider the relation “is an enemy of” and suppose thatthe relation is symmetric and satisfies the condition that for any country, any enemy of an enemy of the country isnot itself an enemy of the country. This is an example of an antitransitive relation that does not have any cycles. Inparticular, by virtue of being antitransitive the relation is not transitive.Finally, let us work with the example of rock, paper, scissors, calling the three options A, B, and C. Now, the relationover A, B, and C is “defeats” and the standard rules of the game are such that A defeats B, B defeats C, and C defeatsA. Furthermore, it is also true that B does not defeat A, C does not defeat B, and A does not defeat C. Finally, it isalso true that no option defeats itself. This information can be depicted in a table:The first argument of the relation is a row and the second one is a column. Ones indicate the relation holds, zeroindicates that it does not hold. Now, notice that the following statement is true for any pair of elements x and y drawn(with replacement) from the set {A, B, C}: If x defeats y, and y defeats z, then x does not defeat z. Hence the relationis antitransitive.Thus, a cycle is neither necessary nor sufficient for a binary relation to be antitransitive.

38.4 Occurrences in preferences• Intransitivity can occur under majority rule, in probabilistic outcomes of game theory, and in the Condorcet

voting method in which ranking several candidates can produce a loop of preference when the weights arecompared (see voting paradox). Intransitive dice demonstrate that probabilities are not necessarily transitive.

• In psychology, intransitivity often occurs in a person’s system of values (or preferences, or tastes), potentiallyleading to unresolvable conflicts.

• Analogously, in economics intransitivity can occur in a consumer’s preferences. This may lead to consumerbehaviour that does not conform to perfect economic rationality. In recent years, economists and philosophershave questioned whether violations of transitivity must necessarily lead to 'irrational behaviour' (see Anand(1993)).

38.5 Likelihood

It has been suggested that Condorcet voting tends to eliminate “intransitive loops” when large numbers of votersparticipate because the overall assessment criteria for voters balances out. For instance, voters may prefer candidateson several different units of measure such as by order of social consciousness or by order of most fiscally conservative.In such cases intransitivity reduces to a broader equation of numbers of people and the weights of their units ofmeasure in assessing candidates.Such as:

https://en.wikipedia.org/wiki/Rock,_paper,_scissors

https://en.wikipedia.org/wiki/Nontransitive_dice

https://en.wikipedia.org/wiki/Penney%2527s_game


https://en.wikipedia.org/wiki/Cycle_(graph_theory)


https://en.wikipedia.org/wiki/Rock,_paper,_scissors

https://en.wikipedia.org/wiki/Majority_rule

https://en.wikipedia.org/wiki/Game_theory

https://en.wikipedia.org/wiki/Condorcet_voting

https://en.wikipedia.org/wiki/Condorcet_voting

https://en.wikipedia.org/wiki/Voting_paradox

https://en.wikipedia.org/wiki/Intransitive_dice

https://en.wikipedia.org/wiki/Psychology

https://en.wikipedia.org/wiki/Value_system


https://en.wikipedia.org/wiki/Taste_(sociology)

https://en.wikipedia.org/wiki/Economics

https://en.wikipedia.org/wiki/Preference#Preference_in_economics

https://en.wikipedia.org/wiki/Rationality#Rationality_in_the_humanities_and_social_sciences

https://en.wikipedia.org/wiki/Condorcet_method

136 CHAPTER 38. INTRANSITIVITY

• 30% favor 60/40 weighting between social consciousness and fiscal conservatism

• 50% favor 50/50 weighting between social consciousness and fiscal conservatism

• 20% favor a 40/60 weighting between social consciousness and fiscal conservatism

While each voter may not assess the units of measure identically, the trend then becomes a single vector on whichthe consensus agrees is a preferred balance of candidate criteria.

38.6 References[1] Wolves do eat grass - see Engel, Cindy (2003). Wild Health: Lessons in Natural Wellness from the Animal Kingdom

(paperback ed.). Houghton Mifflin. p. 141. ISBN 0-618-34068-8..

[2] Guide to Logic, Relations II

[3] IntransitiveRelation

38.7 Further reading• Anand, P (1993). Foundations of Rational Choice Under Risk. Oxford: Oxford University Press..

https://en.wikipedia.org/wiki/Probability_vector

https://en.wikipedia.org/wiki/Consensus



http://www.jgsee.kmutt.ac.th/exell/Logic/Logic42.htm#33

http://www.virtual.cvut.cz/kifb/en/concepts/_intransitive_relation.html

Chapter 39

Inverse relation

For inverse relationships in statistics, see negative relationship.

In mathematics, the inverse relation of a binary relation is the relation that occurs when the order of the elements isswitched in the relation. For example, the inverse of the relation 'child of' is the relation 'parent of'. In formal terms,if X and Y are sets and L ⊆ X × Y is a relation from X to Y then L−1 is the relation defined so that y L−1 x if andonly if xLy . In set-builder notation, L−1 = {(y, x) ∈ Y ×X | (x, y) ∈ L} .The notation comes by analogy with that for an inverse function. Although many functions do not have an inverse;every relation does have a unique inverse. Despite the notation and terminology, the inverse relation is not an inversein the sense of group inverse; the unary operation that maps a relation to the inverse relation is however an involution,so it induces the structure of a semigroup with involution on the binary relations on a set, or more generally induces adagger category on the category of relations as detailed below. As a unary operation, taking the inverse (sometimescalled inversion) commutes however with the order-related operations of relation algebra, i.e. it commutes withunion, intersection, complement etc.The inverse relation is also called the converse relation or transpose relation— the latter in view of its similaritywith the transpose of a matrix.[1] It has also been called the opposite or dual of the original relation.[2] Other notationsfor the inverse relation include LC , LT , L~ or L̆ or L° or L∨.

39.1 Examples

For usual (maybe strict or partial) order relations, the converse is the naively expected “opposite” order, e.g. ≤−1=≥, <−1= > , etc.

39.1.1 Inverse relation of a function

A function is invertible if and only if its inverse relation is a function, in which case the inverse relation is the inversefunction.The inverse relation of a function f : X → Y is the relation f−1 : Y → X defined by graph f−1 = {(y, x) | y =f(x)} .This is not necessarily a function: One necessary condition is that f be injective, since else f−1 is multi-valued. Thiscondition is sufficient for f−1 being a partial function, and it is clear that f−1 then is a (total) function if and only iff is surjective. In that case, i.e. if f is bijective, f−1 may be called the inverse function of f.

39.2 Properties

In the monoid of binary endorelations on a set (with the binary operation on relations being the composition ofrelations), the inverse relation does not satisfy the definition of an inverse from group theory, i.e. if L is an arbitrary

137

https://en.wikipedia.org/wiki/Negative_relationship



https://en.wikipedia.org/wiki/Set-builder_notation


https://en.wikipedia.org/wiki/Group_(mathematics)#Definition




https://en.wikipedia.org/wiki/Dagger_category


https://en.wikipedia.org/wiki/Inverse_relation#Properties



https://en.wikipedia.org/wiki/Transpose





https://en.wikipedia.org/wiki/Multi-valued







https://en.wikipedia.org/wiki/Endorelation




138 CHAPTER 39. INVERSE RELATION

relation on X, then L ◦L−1 does not equal the identity relation on X in general. The inverse relation does satisfy the(weaker) axioms of a semigroup with involution: (L−1)−1 = L and (L ◦R)−1 = R−1 ◦ L−1 .[3]

Since one may generally consider relations between different sets (which form a category rather than a monoid,namely the category of relations Rel), in this context the inverse relation conforms to the axioms of a dagger category(aka category with involution).[3] A relation equal to its inverse is a symmetric relation; in the language of daggercategories, it is self-adjoint.Furthermore, the semigroup of endorelations on a set is also a partially ordered structure (with inclusion of relationsas sets), and actually an involutive quantale. Similarly, the category of heterogenous relations, Rel is also an orderedcategory.[3]

In relation algebra (which is an abstraction of the properties of the algebra of endorelations on a set), inversion (theoperation of taking the inverse relation) commutes with other binary operations of union and intersection. Inversionalso commutes with unary operation of complementation as well as with taking suprema and infima. Inversion is alsocompatible with the ordering of relations by inclusion.[1]

If a relation is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, total, trichotomous, a partialorder, total order, strict weak order, total preorder (weak order), or an equivalence relation, its inverse is too.

39.3 See also• Bijection

• Function (mathematics)

• Inverse function

• Relation (mathematics)

• Transpose graph

39.4 References[1] Gunther Schmidt; Thomas Ströhlein (1993). Relations and Graphs: Discrete Mathematics for Computer Scientists. Springer

Berlin Heidelberg. pp. 9–10. ISBN 978-3-642-77970-1.

[2] Celestina Cotti Ferrero; Giovanni Ferrero (2002). Nearrings: Some Developments Linked to Semigroups and Groups.Kluwer Academic Publishers. p. 3. ISBN 978-1-4613-0267-4.

[3] Joachim Lambek (2001). “Relations Old and New”. In Ewa Orlowska, Andrzej Szalas. Relational Methods for ComputerScience Applications. Springer Science & Business Media. pp. 135–146. ISBN 978-3-7908-1365-4.

• Halmos, Paul R. (1974), Naive Set Theory, p. 40, ISBN 978-0-387-90092-6





https://en.wikipedia.org/wiki/Dagger_category


https://en.wikipedia.org/wiki/Self-adjoint

https://en.wikipedia.org/wiki/Quantale

https://en.wikipedia.org/wiki/Ordered_category

https://en.wikipedia.org/wiki/Ordered_category






















https://en.wikipedia.org/wiki/Transpose_graph





https://en.wikipedia.org/wiki/Joachim_Lambek



https://en.wikipedia.org/wiki/Paul_R._Halmos

https://en.wikipedia.org/wiki/Naive_Set_Theory_(book)



Chapter 40

Inverse trigonometric functions

In mathematics, the inverse trigonometric functions (occasionally called cyclometric functions[1]) are the inversefunctions of the trigonometric functions (with suitably restricted domains). Specifically, they are the inverses of thesine, cosine, tangent, cotangent, secant, and cosecant functions. They are used to obtain an angle from any of theangle’s trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, andgeometry.

40.1 Notation

There are many notations used for the inverse trigonometric functions. The notations sin−1 (x), cos−1 (x), tan−1

(x), etc. are often used, but this convention logically conflicts with the common semantics for expressions like sin2(x), which refer to numeric power rather than function composition, and therefore may result in confusion betweenmultiplicative inverse and compositional inverse. The confusion is somewhat ameliorated by the fact that each of thereciprocal trigonometric functions has its own name—for example, (cos(x))−1=sec(x). Another convention used bysome authors[2] is to use a majuscule (capital/upper-case) first letter along with a −1 superscript, e.g., Sin−1 (x), Cos−1

(x), etc., which avoids confusing them with the multiplicative inverse, which should be represented by sin−1 (x), cos−1

(x), etc. Yet another convention is to use an arc- prefix, so that the confusion with the −1 superscript is resolvedcompletely, e.g., arcsin (x), arccos (x), etc. This convention is used throughout the article. In computer programminglanguages (also MS Office Excel) the inverse trigonometric functions are usually called asin, acos, atan.According to Cajori,[3] the notation sin−1 (x) was introduced by John Herschel in 1813.[4]

40.1.1 Etymology of the arc- prefix

When measuring in radians, an angle of θ radians will correspond to an arc whose length is rθ, where r is the radiusof the circle. Thus, in the unit circle, “the arc whose cosine is x” is the same as “the angle whose cosine is x”, becausethe length of the arc of the circle in radii is the same as the measurement of the angle in radians.[5]

40.2 Basic properties

40.2.1 Principal values

Since none of the six trigonometric functions are one-to-one, they are restricted in order to have inverse functions.Therefore the ranges of the inverse functions are proper subsets of the domains of the original functionsFor example, using function in the sense of multivalued functions, just as the square root function y = √x could bedefined from y2 = x, the function y = arcsin(x) is defined so that sin(y) = x. There are multiple numbers y such thatsin(y) = x; for example, sin(0) = 0, but also sin(π) = 0, sin(2π) = 0, etc. When only one value is desired, the functionmay be restricted to its principal branch. With this restriction, for each x in the domain the expression arcsin(x)

139




https://en.wikipedia.org/wiki/Trigonometric_functions


https://en.wikipedia.org/wiki/Sine

https://en.wikipedia.org/wiki/Cosine

https://en.wikipedia.org/wiki/Trigonometric_functions#Sine.2C_cosine_and_tangent

https://en.wikipedia.org/wiki/Cotangent

https://en.wikipedia.org/wiki/Cotangent#Reciprocal_functions

https://en.wikipedia.org/wiki/Cosecant

https://en.wikipedia.org/wiki/Engineering

https://en.wikipedia.org/wiki/Navigation

https://en.wikipedia.org/wiki/Physics


https://en.wikipedia.org/wiki/Multiplicative_inverse


https://en.wikipedia.org/wiki/Majuscule

https://en.wikipedia.org/wiki/Florian_Cajori

https://en.wikipedia.org/wiki/John_Herschel

https://en.wikipedia.org/wiki/Unit_circle

https://en.wikipedia.org/wiki/One-to-one_function




https://en.wikipedia.org/wiki/Square_root

https://en.wikipedia.org/wiki/Principal_branch

140 CHAPTER 40. INVERSE TRIGONOMETRIC FUNCTIONS

will evaluate only to a single value, called its principal value. These properties apply to all the inverse trigonometricfunctions.The principal inverses are listed in the following table.(Note: Some authors define the range of arcsecant to be ( 0 ≤ y < π/2 or π ≤ y < 3π/2 ), because the tangentfunction is nonnegative on this domain. This makes some computations more consistent. For example using thisrange, tan(arcsec(x))=√x2−1, whereas with the range ( 0 ≤ y < π/2 or π/2 < y ≤ π ), we would have to writetan(arcsec(x))=±√x2−1, since tangent is nonnegative on 0 ≤ y < π/2 but nonpositive on π/2 < y ≤ π. For a similarreason, the same authors define the range of arccosecant to be ( -π < y ≤ -π/2 or 0 < y ≤ π/2 ).)If x is allowed to be a complex number, then the range of y applies only to its real part.

40.2.2 Relationships between trigonometric functions and inverse trigonometric func-tions

Trigonometric functions of inverse trigonometric functions are tabulated below. A quick way to derive them is byconsidering the geometry of a right-angled triangle, with one side of length 1, and another side of length x (any realnumber between 0 and 1), then applying the Pythagorean theorem and definitions of the trigonometric ratios. Purelyalgebraic derivations are longer.

40.2.3 Relationships among the inverse trigonometric functions

Complementary angles:

arccosx =π

2− arcsinx

arccotx =π

2− arctanx

arccscx =π

2− arcsecx

Negative arguments:

arcsin(−x) = − arcsinxarccos(−x) = π − arccosxarctan(−x) = − arctanxarccot(−x) = π − arccotxarcsec(−x) = π − arcsecxarccsc(−x) = − arccscx

Reciprocal arguments:

https://en.wikipedia.org/wiki/Principal_value


https://en.wikipedia.org/wiki/Pythagorean_theorem

40.3. IN CALCULUS 141

arccos(1/x) = arcsecxarcsin(1/x) = arccscx

arctan(1/x) = π

2− arctanx = arccotx , if x > 0

arctan(1/x) = −π2− arctanx = arccotx− π , if x < 0

arccot(1/x) = π

2− arccotx = arctanx , if x > 0

arccot(1/x) = 3π

2− arccotx = π + arctanx , if x < 0

arcsec(1/x) = arccosxarccsc(1/x) = arcsinx

If you only have a fragment of a sine table:

arccosx = arcsin√1− x2 , if 0 ≤ x ≤ 1

arctanx = arcsin x√x2 + 1

Whenever the square root of a complex number is used here, we choose the root with the positive real part (or positiveimaginary part if the square was negative real).From the half-angle formula, tan θ

2 = sin θ1+cos θ , we get:

arcsinx = 2 arctan x

1 +√1− x2

arccosx = 2 arctan√1− x2

1 + x, if − 1 < x ≤ +1

arctanx = 2 arctan x

1 +√1 + x2

40.2.4 Arctangent addition formula

arctanu+ arctan v = arctan(u+ v

1− uv

)(mod π) , uv ̸= 1 .

This is derived from the tangent addition formula

tan(α+ β) =tanα+ tanβ1− tanα tanβ ,

by letting

α = arctanu , β = arctan v .

40.3 In calculus

40.3.1 Derivatives of inverse trigonometric functions

Main article: Differentiation of trigonometric functions

https://en.wikipedia.org/wiki/Tangent_half-angle_formula

https://en.wikipedia.org/wiki/Angle_sum_and_difference_identities

https://en.wikipedia.org/wiki/Differentiation_of_trigonometric_functions


The derivatives for complex values of z are as follows:

ddz arcsin z = 1√

1− z2; z ̸= −1,+1

ddz arccos z = − 1√

1− z2; z ̸= −1,+1

ddz arctan z = 1

1 + z2; z ̸= −i,+i

ddz arccot z = − 1

1 + z2; z ̸= −i,+i

ddz arcsec z = 1

z2√1− 1/z2

; z ̸= −1, 0,+1

ddz arccsc z = − 1

z2√1− 1/z2

; z ̸= −1, 0,+1

Only for real values of x:

ddx arcsecx =

1

|x|√x2 − 1

; |x| > 1

ddx arccscx = − 1

|x|√x2 − 1

; |x| > 1

For a sample derivation: if θ = arcsinx , we get:

d arcsinxdx =

dθd sin θ =

dθcos θdθ =

1

cos θ =1√

1− sin2 θ=

1√1− x2

40.3.2 Expression as definite integrals

Integrating the derivative and fixing the value at one point gives an expression for the inverse trigonometric functionas a definite integral:

arcsinx =

∫ x

0

1√1− z2

dz , |x| ≤ 1

arccosx =

∫ 1

x

1√1− z2

dz , |x| ≤ 1

arctanx =

∫ x

0

1

z2 + 1dz ,

arccotx =

∫ ∞

x

1

z2 + 1dz ,

arcsecx =

∫ x

1

1

z√z2 − 1

dz = π +

∫ −1

x

1

z√z2 − 1

dz , x ≥ 1

arccscx =

∫ ∞

x

1

z√z2 − 1

dz =∫ x

−∞

1

z√z2 − 1

dz , x ≥ 1

When x equals 1, the integrals with limited domains are improper integrals, but still well-defined.

40.3.3 Infinite series

Like the sine and cosine functions, the inverse trigonometric functions can be calculated using power series, as follows.For arcsine, the series can be derived by expanding its derivative, 1√

1−z2 , as a binomial series, and integrating term

https://en.wikipedia.org/wiki/Derivative

https://en.wikipedia.org/wiki/Improper_integral

https://en.wikipedia.org/wiki/Power_series

https://en.wikipedia.org/wiki/Binomial_series

40.3. IN CALCULUS 143

by term (using the integral definition as above). The series for arctangent can similarly be derived by expanding itsderivative 1

1+z2 in a geometric series and applying the integral definition above (see Leibniz series).

arcsin z = z +

(1

2

)z3

3+

(1 · 32 · 4

)z5

5+

(1 · 3 · 52 · 4 · 6

)z7

7+ · · · =

∞∑n=0

(2nn

)z2n+1

4n(2n+ 1); |z| ≤ 1

arccos z = π

2− arcsin z = π

2−(z +

(1

2

)z3

3+

(1 · 32 · 4

)z5

5+ · · ·

)=π

2−

∞∑n=0

(2nn

)z2n+1

4n(2n+ 1); |z| ≤ 1

arctan z = z − z3

3+z5

5− z7

7+ · · · =

∞∑n=0

(−1)nz2n+1

2n+ 1; |z| ≤ 1 z ̸= i,−i

arccot z = π

2−arctan z = π

2−(z − z3

3+z5

5− z7

7+ · · ·

)=π

2−

∞∑n=0

(−1)nz2n+1

2n+ 1; |z| ≤ 1 z ̸= i,−i

arcsec z = arccos(1/z) = π

2−(z−1 +

(1

2

)z−3

3+

(1 · 32 · 4

)z−5

5+ · · ·

)=π

2−

∞∑n=0

(2nn

)z−(2n+1)

4n(2n+ 1); |z| ≥ 1

arccsc z = arcsin(1/z) = z−1 +

(1

2

)z−3

3+

(1 · 32 · 4

)z−5

5+ · · · =

∞∑n=0

(2nn

)z−(2n+1)

4n(2n+ 1); |z| ≥ 1

Leonhard Euler found a more efficient series for the arctangent, which is:

arctan z = z

1 + z2

∞∑n=0

n∏k=1

2kz2

(2k + 1)(1 + z2).

(Notice that the term in the sum for n = 0 is the empty product which is 1.)Alternatively, this can be expressed:

arctan z =∞∑n=0

22n(n!)2

(2n+ 1)!

z2n+1

(1 + z2)n+1

Variant: Continued fractions for arctangent

Two alternatives to the power series for arctangent are these generalized continued fractions:

arctan z = z

1 +(1z)2

3− 1z2 +(3z)2

5− 3z2 +(5z)2

7− 5z2 +(7z)2

9− 7z2 +. . .

=z

1 +(1z)2

3 +(2z)2

5 +(3z)2

7 +(4z)2

9 +. . .

The second of these is valid in the cut complex plane. There are two cuts, from −i to the point at infinity, going downthe imaginary axis, and from i to the point at infinity, going up the same axis. It works best for real numbers runningfrom −1 to 1. The partial denominators are the odd natural numbers, and the partial numerators (after the first) arejust (nz)2, with each perfect square appearing once. The first was developed by Leonhard Euler; the second by CarlFriedrich Gauss utilizing the Gaussian hypergeometric series.

https://en.wikipedia.org/wiki/Geometric_series

https://en.wikipedia.org/wiki/Leibniz_series

https://en.wikipedia.org/wiki/Leonhard_Euler

https://en.wikipedia.org/wiki/Empty_product

https://en.wikipedia.org/wiki/Generalized_continued_fraction

https://en.wikipedia.org/wiki/Leonhard_Euler

https://en.wikipedia.org/wiki/Carl_Friedrich_Gauss


https://en.wikipedia.org/wiki/Gaussian_hypergeometric_series


40.3.4 Indefinite integrals of inverse trigonometric functions

For real and complex values of z:

∫arcsin z dz = z arcsin z +

√1− z2 + C∫

arccos z dz = z arccos z −√

1− z2 + C∫arctan z dz = z arctan z − 1

2ln(1 + z2

)+ C∫

arccot z dz = z arccot z + 1

2ln(1 + z2

)+ C∫

arcsec z dz = z arcsec z − ln[z

(1 +

√z2 − 1

z2

)]+ C

∫arccsc z dz = z arccsc z + ln

[z

(1 +

√z2 − 1

z2

)]+ C

For real x ≥ 1:

∫arcsecx dx = x arcsecx− ln

(x+

√x2 − 1

)+ C∫

arccscx dx = x arccscx+ ln(x+

√x2 − 1

)+ C

For all real x not between −1 and 1:

∫arcsecx dx = x arcsecx− sgn(x) ln

∣∣∣x+√x2 − 1

∣∣∣+ C∫arccscx dx = x arccscx+ sgn(x) ln

∣∣∣x+√x2 − 1

∣∣∣+ C

The absolute value is necessary to compensate for both negative and positive values of the arcsecant and arccosecantfunctions. The signum function is also necessary due to the absolute values in the derivatives of the two functions,which create two different solutions for positive and negative values of x. These can be further simplified using thelogarithmic definitions of the inverse hyperbolic functions:

∫arcsecx dx = x arcsecx− arcosh |x|+ C∫arccscx dx = x arccscx+ arcosh |x|+ C

The absolute value in the argument of the arcosh function creates a negative half of its graph, making it identical tothe signum logarithmic function shown above.All of these antiderivatives can be derived using integration by parts and the simple derivative forms shown above.

Example

Using∫u dv = uv −

∫v du , set

u = arcsinx dv = dx

du =dx√1− x2

v = x

https://en.wikipedia.org/wiki/Inverse_trigonometric_functions#Derivatives_of_inverse_trigonometric_functions

https://en.wikipedia.org/wiki/Inverse_hyperbolic_functions#Logarithmic_representation

https://en.wikipedia.org/wiki/Integration_by_parts

40.4. EXTENSION TO COMPLEX PLANE 145

Then

∫arcsin(x) dx = x arcsinx−

∫x√

1− x2dx

Substitute

w = 1− x2 .

Then

dw = −2x dx

and

∫x√

1− x2dx = −1

2

∫ dw√w

= −√w

Back-substitute for x to yield

∫arcsin(x) dx = x arcsinx+

√1− x2 + C

40.4 Extension to complex plane

Since the inverse trigonometric functions are analytic functions, they can be extended from the real line to the complexplane. This results in functions with multiple sheets and branch points. One possible way of defining the extensionsis:

arctan z =∫ z

0

dx1 + x2

z ̸= −i,+i

where the part of the imaginary axis which does not lie strictly between −i and +i is the cut between the principalsheet and other sheets;

arcsin z = arctan z√1− z2

z ̸= −1,+1

where (the square-root function has its cut along the negative real axis and) the part of the real axis which does notlie strictly between −1 and +1 is the cut between the principal sheet of arcsin and other sheets;

arccos z = π

2− arcsin z z ̸= −1,+1

which has the same cut as arcsin;

arccot z = π

2− arctan z z ̸= −i,+i

which has the same cut as arctan;

https://en.wikipedia.org/wiki/Integration_by_substitution

https://en.wikipedia.org/wiki/Analytic_function

https://en.wikipedia.org/wiki/Branch_point


arcsec z = arccos 1z

z ̸= −1, 0,+1

where the part of the real axis between −1 and +1 inclusive is the cut between the principal sheet of arcsec and othersheets;

arccsc z = arcsin 1

zz ̸= −1, 0,+1

which has the same cut as arcsec.

40.4.1 Logarithmic forms

These functions may also be expressed using complex logarithms. This extends in a natural fashion their domain tothe complex plane.

arcsin z = −i ln(iz +

√1− z2

)= arccsc 1

z

arccos z = −i ln(z +

√z2 − 1

)=π

2+ i ln

(iz +

√1− z2

)=π

2− arcsin z = arcsec 1

z

arctan z = 12 i [ln (1− iz)− ln (1 + iz)] = arccot 1

z

arccot z = 12 i

[ln(1− i

z

)− ln

(1 +

i

z

)]= arctan 1

z

arcsec z = −i ln(√

1

z2− 1 +

1

z

)= i ln

(√1− 1

z2+i

z

)+π

2=π

2− arccsc z = arccos 1

z

arccsc z = −i ln(√

1− 1

z2+i

z

)= arcsin 1

z

Elementary proofs of these relations proceed via expansion to exponential forms of the trigonometric functions.

Example proof

sin(ϕ) = z

ϕ = arcsin z

Using the exponential definition of sine, one obtains

z =eiϕ − e−iϕ

2i

Let

ξ = eiϕ

Solving for ϕ

https://en.wikipedia.org/wiki/Complex_logarithm


https://en.wikipedia.org/wiki/Complex_plane

https://en.wikipedia.org/wiki/Trigonometric_function#Relationship_to_exponential_function_and_complex_numbers

40.5. APPLICATIONS 147

z =ξ − 1/ξ

2i

2iz = ξ − 1/ξ

ξ − 2iz − 1/ξ = 0

ξ2 − 2iξz − 1 = 0

ξ = iz ±√1− z2 = eiϕ

iϕ = ln(iz ±

√1− z2

)ϕ = −i ln

(iz ±

√1− z2

)(the positive branch is chosen)

ϕ = arcsin z = −i ln(iz +

√1− z2

)Example proof (variant 2)

ϕ = arcsin zeiϕ = cos(ϕ) + i sin(ϕ)Apply the natural logarithm, multiply by -i and substitute phi.arcsin z = −i ln[cos(arcsin z) + i sin(arcsin z)]arcsin z = −i ln(

√1− z2 + iz)

40.5 Applications

40.5.1 General solutions

Each of the trigonometric functions is periodic in the real part of its argument, running through all its values twice ineach interval of 2π. Sine and cosecant begin their period at 2πk − π/2 (where k is an integer), finish it at 2πk + π/2,and then reverse themselves over 2πk + π/2 to 2πk + 3π/2. Cosine and secant begin their period at 2πk, finish it at2πk + π, and then reverse themselves over 2πk + π to 2πk + 2π. Tangent begins its period at 2πk − π/2, finishes it at2πk + π/2, and then repeats it (forward) over 2πk + π/2 to 2πk + 3π/2. Cotangent begins its period at 2πk, finishesit at 2πk + π, and then repeats it (forward) over 2πk + π to 2πk + 2π.This periodicity is reflected in the general inverses where k is some integer:

sin(y) = x ⇔ y = arcsin(x) + 2πk or y = π − arcsin(x) + 2πk

sin(y) = x ⇔ y = (−1)k arcsin(x) + πk

cos(y) = x ⇔ y = arccos(x) + 2πk or y = 2π − arccos(x) + 2πk

cos(y) = x ⇔ y = ± arccos(x) + 2πk

tan(y) = x ⇔ y = arctan(x) + πk

cot(y) = x ⇔ y = arccot(x) + πk

sec(y) = x ⇔ y = arcsec(x) + 2πk or y = 2π − arcsec(x) + 2πk

csc(y) = x ⇔ y = arccsc(x) + 2πk or y = π − arccsc(x) + 2πk


Application: finding the angle of a right triangle

Inverse trigonometric functions are useful when trying to determine the remaining two angles of a right triangle whenthe lengths of the sides of the triangle are known. Recalling the right-triangle definitions of sine, for example, itfollows that

θ = arcsin( opposite

hypotenuse

).

Often, the hypotenuse is unknown and would need to be calculated before using arcsine or arccosine using thePythagorean Theorem: a2 + b2 = h2 where h is the length of the hypotenuse. Arctangent comes in handy inthis situation, as the length of the hypotenuse is not needed.

θ = arctan(opposite

adjacent

).

For example, suppose a roof drops 8 feet as it runs out 20 feet. The roof makes an angle θ with the horizontal, whereθ may be computed as follows:

θ = arctan(opposite

adjacent

)= arctan

( riserun

)= arctan

(8

20

)≈ 21.8◦ .

40.5.2 In computer science and engineering

Two-argument variant of arctangent

Main article: atan2

The two-argument atan2 function computes the arctangent of y / x given y and x, but with a range of (−π, π]. Inother words, atan2(y, x) is the angle between the positive x-axis of a plane and the point (x, y) on it, with positivesign for counter-clockwise angles (upper half-plane, y > 0), and negative sign for clockwise angles (lower half-plane,y < 0). It was first introduced in many computer programming languages, but it is now also common in other fieldsof science and engineering.In terms of the standard arctan function, that is with range of (−π/2, π/2), it can be expressed as follows:

atan2(y, x) =

arctan( yx ) x > 0

arctan( yx ) + π y ≥ 0 , x < 0

arctan( yx )− π y < 0 , x < 0π2 y > 0 , x = 0

−π2 y < 0 , x = 0

undefined y = 0 , x = 0

It also equals the principal value of the argument of the complex number x + iy.This function may also be defined using the tangent half-angle formulae as follows:

atan2(y, x) = 2 arctan y√x2 + y2 + x

provided that either x > 0 or y ≠ 0. However this fails if given x ≤ 0 and y = 0 so the expression is unsuitable forcomputational use.The above argument order (y, x) seems to be the most common, and in particular is used in ISO standards such asthe C programming language, but a few authors may use the opposite convention (x, y) so some caution is warranted.These variations are detailed at atan2.

https://en.wikipedia.org/wiki/Right_triangle

https://en.wikipedia.org/wiki/Pythagorean_Theorem

https://en.wikipedia.org/wiki/Atan2


https://en.wikipedia.org/wiki/Principal_value

https://en.wikipedia.org/wiki/Arg_(mathematics)



https://en.wikipedia.org/wiki/ISO_standard

https://en.wikipedia.org/wiki/C_(programming_language)

https://en.wikipedia.org/wiki/Atan2#Variations_and_notation

40.6. SEE ALSO 149

Arctangent function with location parameter

In many applications the solution y of the equation x = tan y is to come as close as possible to a given value−∞ < η <∞ . The adequate solution is produced by the parameter modified arctangent function

y = arctanη x := arctanx+ π · rni η − arctanxπ

.

The function rni rounds to the nearest integer.

Practical considerations

For angles near 0 and π, arccosine is ill-conditioned and will thus calculate the angle with reduced accuracy in acomputer implementation (due to the limited number of digits). Similarly, arcsine is inaccurate for angles near −π/2and π/2. To achieve full accuracy for all angles, arctangent or atan2 should be used for the implementation.

40.6 See also• Argument (complex analysis)

• Complex logarithm

• Gauss’s continued fraction

• Inverse hyperbolic function

• List of integrals of inverse trigonometric functions

• List of trigonometric identities

• Square root

• Tangent half-angle formula

• Trigonometric function

40.7 References[1] For example Dörrie, Heinrich (1965). Triumph der Mathematik. Trans. David Antin. Dover. p. 69. ISBN 0-486-61348-8.

[2] Prof. Sanaullah Bhatti; Ch. Nawab-ud-Din; Ch. Bashir Ahmed; Dr. S. M. Yousuf; Dr. Allah Bukhsh Taheem (1999).“Differentiation of Tigonometric, Logarithmic and Exponential Functions”. In Prof. Mohammad Maqbool Ellahi, Dr.Karamat Hussain Dar, Faheem Hussain. Calculus and Analytic Geometry (in Pakistani English) (First ed.). Lahore: PunjabTextbook Board. p. 140.

[3] Cajori, Florian (1919). A History of Mathematics (2nd ed.). The Macmillan Company, New York. p. 272., at GoogleBooks

[4] Herschel, John F. W. (1813). “On a remarkable Application of Cotes’s Theorem”. Philosophical Transactions (RoyalSociety, London) 103 (1): 10., at Google Books

[5] “Inverse trigonometric functions” in The Americana: a universal reference library, Vol.21, Ed. Frederick Converse Beach,George Edwin Rines, (1912).

40.8 External links• Weisstein, Eric W., “Inverse Trigonometric Functions”, MathWorld.

• Weisstein, Eric W., “Inverse Tangent”, MathWorld.

https://en.wikipedia.org/wiki/Ill-conditioned


https://en.wikipedia.org/wiki/Argument_(complex_analysis)

https://en.wikipedia.org/wiki/Complex_logarithm

https://en.wikipedia.org/wiki/Gauss%2527s_continued_fraction

https://en.wikipedia.org/wiki/Inverse_hyperbolic_function

https://en.wikipedia.org/wiki/List_of_integrals_of_inverse_trigonometric_functions

https://en.wikipedia.org/wiki/List_of_trigonometric_identities



https://en.wikipedia.org/wiki/Trigonometric_function



https://en.wikipedia.org/wiki/Lahore

https://books.google.com/books?id=bBoPAAAAIAAJ

https://books.google.com/books


https://books.google.com/books?id=qpRJAAAAYAAJ



http://mathworld.wolfram.com/InverseTrigonometricFunctions.html



http://mathworld.wolfram.com/InverseTangent.html



The usual principal values of the arcsin(x) (red) and arccos(x) (blue) functions graphed on the cartesian plane.


The usual principal values of the arctan(x) and arccot(x) functions graphed on the cartesian plane.

Principal values of the arcsec(x) and arccsc(x) functions graphed on the cartesian plane.


A C

B

b

ah

(adjacent)

(opposite)(hypotenuse)

A right triangle.

Chapter 41

Near sets

Figure 1. Descriptively, very near sets

In mathematics, near sets are either spatially close or descriptively close. Spatially close sets have nonempty intersection.In other words, spatially close sets are not disjoint sets, since they always have at least one element in common. De-scriptively close sets contain elements that have matching descriptions. Such sets can be either disjoint or non-disjointsets. Spatially near sets are also descriptively near sets.The underlying assumption with descriptively close sets is that such sets contain elements that have location andmeasurable features such as colour and frequency of occurrence. The description of the element of a set is definedby a feature vector. Comparison of feature vectors provides a basis for measuring the closeness of descriptively near

153

https://en.wikipedia.org/wiki/Closeness_(mathematics)




https://en.wikipedia.org/wiki/Feature_vector

154 CHAPTER 41. NEAR SETS

Figure 2. Descriptively, minimally near sets

sets. Near set theory provides a formal basis for the observation, comparison, and classification of elements in setsbased on their closeness, either spatially or descriptively. Near sets offer a framework for solving problems based onhuman perception that arise in areas such as image processing, computer vision as well as engineering and scienceproblems.Near sets have a variety of applications in areas such as topology[37], pattern detection and classification[50], abstract al-gebra[51], mathematics in computer science[38], and solving a variety of problems based on human perception[42][82][47][52][56]

that arise in areas such as image analysis[54][14][46][17][18], image processing[40], face recognition[13], ethology[64], aswell as engineering and science problems[55][64][42][19][17][18]. From the beginning, descriptively near sets have provedto be useful in applications of topology[37], and visual pattern recognition [50], spanning a broad spectrum of applica-tions that include camouflage detection, micropaleontology, handwriting forgery detection, biomedical image analy-sis, content-based image retrieval, population dynamics, quotient topology, textile design, visual merchandising, andtopological psychology.As an illustration of the degree of descriptive nearness between two sets, consider an example of the Henry colourmodel for varying degrees of nearness between sets of picture elements in pictures (see, e.g.,[17] §4.3). The two pairsof ovals in Fig. 1 and Fig. 2 contain coloured segments. Each segment in the figures corresponds to an equivalenceclass where all pixels in the class have similar descriptions, i.e., picture elements with similar colours. The ovals inFig.1 are closer to each other descriptively than the ovals in Fig. 2.

https://en.wikipedia.org/wiki/Perception

https://en.wikipedia.org/wiki/Image_processing

https://en.wikipedia.org/wiki/Computer_vision


https://en.wikipedia.org/wiki/Near_sets#cnote_37

https://en.wikipedia.org/wiki/Pattern_recognition

https://en.wikipedia.org/wiki/Statistical_classification











https://en.wikipedia.org/wiki/Image_analysis







https://en.wikipedia.org/wiki/Facial_recognition_system


https://en.wikipedia.org/wiki/Ethology










https://en.wikipedia.org/wiki/Camouflage

https://en.wikipedia.org/wiki/Micropaleontology

https://en.wikipedia.org/wiki/Content-based_image_retrieval

https://en.wikipedia.org/wiki/Population_dynamics


https://en.wikipedia.org/wiki/Textile_design

https://en.wikipedia.org/wiki/Visual_merchandising


41.1. HISTORY 155

41.1 History

It has been observed that the simple concept of nearness unifies various concepts of topological structures[20] inas-much as the category Near of all nearness spaces and nearness preserving maps contains categories sTop (symmetrictopological spaces and continuous maps[3]),Prox (proximity spaces and δ -maps[8][67]),Unif (uniform spaces and uni-formly continuous maps[81][77]) andCont (contiguity spaces and contiguity maps[24]) as embedded full subcategories[20][59].The categories εANear and εAMer are shown to be full supercategories of various well-known categories, in-cluding the category sTop of symmetric topological spaces and continuous maps, and the category Met∞ ofextended metric spaces and nonexpansive maps. The notation A ↪→ B reads category A is embedded in category B. The categories εAMer and εANear are supercategories for a variety of familiar categories[76] shown in Fig. 3.Let εANear denote the category of all ε -approach nearness spaces and contractions, and let εAMer denote thecategory of all ε -approach merotopic spaces and contractions.

Figure 3. Supercats

Among these familiar categories is sTop , the symmetric form of Top (see category of topological spaces), the cat-egory with objects that are topological spaces and morphisms that are continuous maps between them[1][32]. Met∞

with objects that are extended metric spaces is a subcategory of εAP (having objects ε -approach spaces and con-tractions) (see also[57][75]). Let ρX , ρY be extended pseudometrics on nonempty sets X,Y , respectively. The mapf : (X, ρX) −→ (Y, ρY ) is a contraction if and only if f : (X, νDρX

) −→ (Y, νDρY) is a contraction. For

nonempty subsets A,B ∈ 2X , the distance function Dρ : 2X × 2X −→ [0,∞] is defined by

Dρ(A,B) =

{inf {ρ(a, b) : a ∈ A, b ∈ B}, ifA and Bempty not are ,∞, ifA or Bempty is .

Thus ε AP is embedded as a full subcategory in εANear by the functor F : εAP −→ εANear defined byF ((X, ρ)) = (X, νDρ) and F (f) = f . Then f : (X, ρX) −→ (Y, ρY ) is a contraction if and only if f :




https://en.wikipedia.org/wiki/Proximity_space



https://en.wikipedia.org/wiki/Uniform_space







https://en.wikipedia.org/wiki/Category_of_topological_spaces






(X, νDρX) −→ (Y, νDρY

) is a contraction. Thus εAP is embedded as a full subcategory in εANear by thefunctor F : εAP −→ εANear defined by F ((X, ρ)) = (X, νDρ) and F (f) = f. Since the category Met∞

of extended metric spaces and nonexpansive maps is a full subcategory of εAP , therefore, εANear is also a fullsupercategory of Met∞ . The category εANear is a topological construct[76].

Figure 4. Frigyes Riesz, 1880-1956

The notions of near and far[A] in mathematics can be traced back to works by Johann Benedict Listing and FelixHausdorff. The related notions of resemblance and similarity can be traced back to J.H. Poincaré, who introducedsets of similar sensations (nascent tolerance classes) to represent the results of G.T. Fechner’s sensation sensitivityexperiments[10] and a framework for the study of resemblance in representative spaces as models of what he termedphysical continua[63][60][61]. The elements of a physical continuum (pc) are sets of sensations. The notion of a pcand various representative spaces (tactile, visual, motor spaces) were introduced by Poincaré in an 1894 article onthe mathematical continuum[63], an 1895 article on space and geometry[60] and a compendious 1902 book on science


https://en.wikipedia.org/wiki/Frigyes_Riesz

https://en.wikipedia.org/wiki/Near_sets#cnote_A_grp_2

https://en.wikipedia.org/wiki/Johann_Benedict_Listing

https://en.wikipedia.org/wiki/Felix_Hausdorff

https://en.wikipedia.org/wiki/Felix_Hausdorff

https://en.wikipedia.org/wiki/Henri_Poincar%C3%A9







41.2. NEARNESS OF SETS 157

and hypothesis[61] followed by a number of elaborations, e.g.,[62]. The 1893 and 1895 articles on continua (Pt. 1,ch. II) as well as representative spaces and geometry (Pt. 2, ch IV) are included as chapters in[61]. Later, F. Rieszintroduced the concept of proximity or nearness of pairs of sets at the International Congress of Mathematicians(ICM) in 1908[65].During the 1960s, E.C. Zeeman introduced tolerance spaces in modelling visual perception[83]. A.B. Sossinskyobserved in 1986[71] that the main idea underlying tolerance space theory comes from Poincaré, especially[60]. In2002, Z. Pawlak and J. Peters[B] considered an informal approach to the perception of the nearness of physical objectssuch as snowflakes that was not limited to spatial nearness. In 2006, a formal approach to the descriptive nearness ofobjects was considered by J. Peters, A. Skowron and J. Stepaniuk[C] in the context of proximity spaces[39][33][35][21].In 2007, descriptively near sets were introduced by J. Peters[D][E] followed by the introduction of tolerance nearsets[41][45]. Recently, the study of descriptively near sets has led to algebraic[22][51], topological and proximity space[37]

foundations of such sets.

41.2 Nearness of sets

The adjective near in the context of near sets is used to denote the fact that observed feature value differences ofdistinct objects are small enough to be considered indistinguishable, i.e., within some tolerance.The exact idea of closeness or 'resemblance' or of 'being within tolerance' is universal enough to appear, quite naturally,in almost any mathematical setting (see, e.g.,[66]). It is especially natural in mathematical applications: practicalproblems, more often than not, deal with approximate input data and only require viable results with a tolerable levelof error[71].The words near and far are used in daily life and it was an incisive suggestion of F. Riesz[65] that these intuitiveconcepts be made rigorous. He introduced the concept of nearness of pairs of sets at the ICM in Rome in 1908. Thisconcept is useful in simplifying teaching calculus and advanced calculus. For example, the passage from an intuitivedefinition of continuity of a function at a point to its rigorous epsilon-delta definition is sometime difficult for teachersto explain and for students to understand. Intuitively, continuity can be explained using nearness language, i.e., afunction f : R → R is continuous at a point c , provided points {x} near c go into points {f(x)} near f(c) . UsingRiesz’s idea, this definition can be made more precise and its contrapositive is the familiar definition[4][36].

41.3 Generalization of set intersection

From a spatial point of view, nearness (aka proximity) is considered a generalization of set intersection. For disjointsets, a form of nearness set intersection is defined in terms of a set of objects (extracted from disjoint sets) that havesimilar features within some tolerance (see, e.g., §3 in[80]). For example, the ovals in Fig. 1 are considered near eachother, since these ovals contain pairs of classes that display similar (visually indistinguishable) colours.

41.4 Efremovič proximity space

Let X denote a metric topological space that is endowed with one or more proximity relations and let 2X denote thecollection of all subsets of X . The collection 2X is called the power set of X .There are many ways to define Efremovič proximities on topological spaces (discrete proximity, standard proximity,metric proximity, Čech proximity, Alexandroff proximity, and Freudenthal proximity), For details, see § 2, pp. 93–94 in[6]. The focus here is on standard proximity on a topological space. For A,B ⊂ X , A is near B (denoted byA δ B ), provided their closures share a common point.The closure of a subset A ∈ 2X (denoted by cl(A) ) is the usual Kuratowski closure of a set[F], introduced in § 4, p.20[27], is defined by

cl(A) = {x ∈ X : D(x,A) = 0} , whereD(x,A) = inf {d(x, a) : a ∈ A} .

i.e. cl(A) is the set of all points x in X that are close to A ( D(x,A) is the Hausdorff distance (see § 22, p. 128,




https://en.wikipedia.org/wiki/International_Congress_of_Mathematicians


https://en.wikipedia.org/wiki/Christopher_Zeeman




https://en.wikipedia.org/wiki/Near_sets#cnote_B_grp_2

https://en.wikipedia.org/wiki/Near_sets#cnote_C_grp_2





https://en.wikipedia.org/wiki/Near_sets#cnote_D_grp_2

https://en.wikipedia.org/wiki/Near_sets#cnote_E_grp_2















https://en.wikipedia.org/wiki/Metric_space




https://en.wikipedia.org/wiki/Kuratowski_closure_axioms

https://en.wikipedia.org/wiki/Near_sets#cnote_F_grp_2



in[15]) between x and the set A and d(x, a) = |x− a| (standard distance)). A standard proximity relation is definedby

δ ={(A,B) ∈ 2X × 2X : cl(A) ∩ cl(B) ̸= ∅

}.

Whenever sets A and B have no points in common, the sets are farfrom each other (denoted A δ B ).The following EF-proximity[G] space axioms are given by Jurij Michailov Smirnov[67] based on what Vadim Arsenye-vič Efremovič introduced during the first half of the 1930s[8]. Let A,B,E ∈ 2X .

EF.1 If the set A is close to B , then B is close to A .

EF.2 A ∪B is close to E , if and only if, at least one of the sets A or B is close to E .

EF.3 Two points are close, if and only if, they are the same point.

EF.4 All sets are far from the empty set ∅ .

EF.5 For any two sets A and B which are far from each other, there exists C,D ∈ 2X , C ∪D = X , such that Ais far from C and B is far from D (Efremovič-axiom).

The pair (X, δ) is called an EF-proximity space. In this context, a space is a set with some added structure. Witha proximity space X , the structure of X is induced by the EF-proximity relation δ . In a proximity space X , theclosure of A in X coincides with the intersection of all closed sets that contain A .

Theorem 1[67] The closure of any set A in the proximity space X is the set of points x ∈ X that are close to A .

41.5 Visualization of EF-axiom

Let the set X be represented by the points inside the rectangular region in Fig. 5. Also, let A,B be any two non-intersection subsets (i.e. subsets spatially far from each other) inX , as shown in Fig. 5. LetCc = X\C (complementof the set C ). Then from the EF-axiom, observe the following:

A δ B,

B ⊂ C,

D = Cc,

X = D ∪ C,A ⊂ D, hence, we can write

A δ B ⇒ A δ C and B δ D, for some C,D in X so that C ∪D = X. ■

41.6 Descriptive proximity space

Descriptively near sets were introduced as a means of solving classification and pattern recognition problems arisingfrom disjoint sets that resemble each other[44][43]. Recently, the connections between near sets in EF-spaces and nearsets in descriptive EF-proximity spaces have been explored in[53][48].Again, let X be a metric topological space and let Φ = {ϕ1, . . . , ϕn} a set of probe functions that represent featuresof each x ∈ X . The assumption made here is X contains non-abstract points that have measurable features such asgradient orientation. A non-abstract point has a location and features that can be measured (see § 3 in [26]).A probe function ϕ : X → R represents a feature of a sample point in X . The mapping Φ : X −→ Rn is definedby Φ(x) = (ϕ1(x), . . . , ϕn(x)) , where Rn is an n-dimensional real Euclidean vector space. Φ(x) is a feature vectorfor x , which provides a description of x ∈ X . For example, this leads to a proximal view of sets of picture pointsin digital images[48].


https://en.wikipedia.org/wiki/Near_sets#cnote_G_grp_2


https://en.wikipedia.org/wiki/Vadim_Arsenyevich_Efremovich




https://en.wikipedia.org/wiki/Space










41.6. DESCRIPTIVE PROXIMITY SPACE 159

X

AB

CC c

Figure 5. Example of a descriptive EF-proximity relation between sets A,B , and Cc

To obtain a descriptive proximity relation (denoted by δΦ ), one first chooses a set of probe functions. LetQ : 2X −→2R

n be a mapping on a subset of 2X into a subset of 2Rn . For example, let A,B ∈ 2X and Q(A),Q(B) denotesets of descriptions of points in A,B , respectively. That is,

Q(A) = {Φ(a) : a ∈ A} ,Q(B) = {Φ(b) : b ∈ B} .

The expression A δΦ B reads A is descriptively near B . Similarly, A δΦ B reads A is descriptively far from B . Thedescriptive proximity of A and B is defined by

A δΦ B ⇔ Q(cl(A)) δ Q(cl(B)) ̸= ∅.

The descriptive intersection ∩Φ of A and B is defined by

A ∩Φ B = {x ∈ A ∪B : Q(A) δ Q(B)} .

That is, x ∈ A ∪B is in A ∩Φ B , provided Φ(x) = Φ(a) = Φ(b) for some a ∈ A, b ∈ B . Observe that A and Bcan be disjoint and yet A ∩Φ B can be nonempty. The descriptive proximity relation δΦ is defined by

δΦ ={(A,B) ∈ 2X × 2X : cl(A) ∩Φ cl(B) ̸= ∅

}.

Whenever sets A and B have no points with matching descriptions, the sets are descriptively far from each other(denoted by A δΦ B ).The binary relation δΦ is a descriptive EF-proximity, provided the following axioms are satisfied for A,B,E ⊂ X .


dEF.1 If the set A is descriptively close to B , then B is descriptively close to A .

dEF.2 A ∪B is descriptively close to E , if and only if, at least one of the sets A or B is descriptively close to E .

dEF.3 Two points x, y ∈ X are descriptively close, if and only if, the description of x matches the description ofy .

dEF.4 All nonempty sets are descriptively far from the empty set ∅ .

dEF.5 For any two setsA andB which are descriptively far from each other, there exists C,D ∈ 2X , C ∪D = X, such that A is descriptively far from C and B is descriptively far from D (Descriptive Efremovič axiom).

The pair (X, δΦ) is called a descriptive proximity space.

41.7 Proximal relator spaces

A relator is a nonvoid family of relations R on a nonempty set X [72]. The pair (X,R) (also denoted X(R) ) iscalled a relator space. Relator spaces are natural generalizations of ordered sets and uniform spaces[73][74]}. With theintroduction of a family of proximity relations Rδ onX , we obtain a proximal relator space (X,Rδ) . For simplicity,we consider only two proximity relations, namely, the Efremovič proximity δ [8] and the descriptive proximity δΦ indefining the descriptive relator RδΦ

[53][48]. The pair (X,RδΦ) is called a proximal relator space [49]. In this work, Xdenotes a metric topological space that is endowed with the relations in a proximal relator. With the introduction of(X,RδΦ) , the traditional closure of a subset (e.g., [9][7]) can be compared with the more recent descriptive closureof a subset.In a proximal relator space X , the descriptive closure of a set A (denoted by clΦ(A) ) is defined by

clΦ(A) = {x ∈ X : Φ(x)δQ(cl(A))} .

That is, x ∈ X is in the descriptive closure of A , provided the closure of Φ(x) and the closure of Q(cl(A)) have atleast one element in common.

Theorem 2 [50] The descriptive closure of any set A in the descriptive EF-proximity space (X,RδΦ) is the set ofpoints x ∈ X that are descriptively close to A .

Theorem 3 [50] Kuratowski closure of a setA is a subset of the descriptive closure ofA in a descriptive EF-proximityspace.

Theorem 4 [49] Let (X,RδΦ) be a proximal relator space, A ⊂ X . Then cl(A) ⊆ clΦ(A) .

Proof Let Φ(x) ∈ Q(X \ cl(A)) such that Φ(x) = Φ(a) for some a ∈ clA . Consequently, Φ(x) ∈ Q(clΦ(A)) .Hence, cl(A) ⊆ clΦ(A)

In a proximal relator space, EF-proximity δ leads to the following results for descriptive proximity δΦ .

Theorem 5 [49] Let (X,RδΦ) be a proximal relator space, A,B,C ⊂ X . Then

1 ◦

A δ B implies A δΦ B .

2 ◦

(A ∪B) δ C implies (A ∪B) δΦ C .

3 ◦

clA δ clB implies clA δΦ clB .

Proof














41.8. DESCRIPTIVE δ -NEIGHBOURHOODS 161

1 ◦

A δ B ⇔ A ∩B ̸= ∅ . For x ∈ A ∩B,Φ(x) ∈ Q(A) and Φ(x) ∈ Q(B) . Consequently, A δΦ B .

1◦ ⇒ 2◦

3 ◦

clA δ clB implies that clA and clA have at least one point in common. Hence, 1 o ⇒ 3o .

■

41.8 Descriptive δ -neighbourhoods

X

BE 1

E 2

A

X \E 2

Figure 6. Example depicting δ -neighbourhoods

In a pseudometric proximal relator space X , the neighbourhood of a point x ∈ X (denoted by Nx,ε ), for ε > 0 , isdefined by

Nx,ε = {y ∈ X : d(x, y) < ε} .

The interior of a set A (denoted by int(A) ) and boundary of A (denoted by bdy(A) ) in a proximal relator space Xare defined by

int(A) = {x ∈ X : Nx,ε ⊆ A} .

bdy(A) = cl(A) \ int(A).


A setA has a natural strong inclusion in a setB associated with δ [5][6]} (denoted byA ≪δ B ), providedA ⊂ intB, i.e., A δ X \ intB ( A is far from the complement of intB ). Correspondingly, a set A has a descriptive stronginclusion in a set B associated with δΦ (denoted by A≪Φ B ), provided Q(A) ⊂ Q(intB) , i.e., A δΦ X \ intB (Q(A) is far from the complement of intB ).Let ≪Φ be a descriptive δ -neighbourhood relation defined by

≪Φ ={(A,B) ∈ 2X × 2X : Q(A) ⊂ Q(intB)

}.

That is,A≪Φ B , provided the description of each a ∈ A is contained in the set of descriptions of the points b ∈ intB. Now observe that any A,B in the proximal relator space X such that A δΦ B have disjoint δΦ -neighbourhoods,i.e.,

A δΦ B ⇔ A≪Φ E1, B ≪Φ E2, for some E1, E2 ⊂ X (See Fig. 6).

Theorem 6 [50] Any two sets descriptively far from each other belong to disjoint descriptive δΦ -neighbourhoods ina descriptive proximity space X .

A consideration of strong containment of a nonempty set in another set leads to the study of hit-and-miss topologiesand the Wijsman topology[2].

41.9 Tolerance near sets

Let ε be a real number greater than zero. In the study of sets that are proximally near within some tolerance, the setof proximity relations RδΦ is augmented with a pseudometric tolerance proximity relation (denoted by δΦ,ε ) definedby

DΦ(A,B) = inf {d(Φ(a),Φ(a)) : Φ(a) ∈ Q(A),Φ(a) ∈ Q(B)} ,

d(Φ(a),Φ(a)) =∑n

i=1|ϕi(a)− ϕi(b)|,

δΦ,ε ={(A,B) ∈ 2X × 2X : |D(cl(A), cl(B))| < ε

}.

Let RδΦ,ε = RδΦ ∪{δΦ,ε} . In other words, a nonempty set equipped with the proximal relator RδΦ,ε has underlyingstructure provided by the proximal relator RδΦ and provides a basis for the study of tolerance near sets in X that arenear within some tolerance. SetsA,B in a descriptive pseudometric proximal relator space (X,RδΦ,ε) are tolerancenear sets (i.e., A δΦ,ε B ), provided

DΦ(A,B) < ε.

41.10 Tolerance classes and preclasses

Relations with the same formal properties as similarity relations of sensations considered by Poincaré[62] are nowadays,after Zeeman[83], called tolerance relations. A tolerance τ on a set O is a relation τ ⊆ O × O that is reflexive andsymmetric. In algebra, the term tolerance relation is also used in a narrow sense to denote reflexive and symmetricrelations defined on universes of algebras that are also compatible with operations of a given algebra, i.e., they aregeneralizations of congruence relations (see e.g.,[12]). In referring to such relations, the term algebraic tolerance orthe term algebraic tolerance relation is used. Transitive tolerance relations are equivalence relations. A setO togetherwith a tolerance τ is called a tolerance space (denoted (O, τ) ). A set A ⊆ O is a τ -preclass (or briefly preclasswhen τ is understood) if and only if for any x, y ∈ A , (x, y) ∈ τ .The family of all preclasses of a tolerance space is naturally ordered by set inclusion and preclasses that are maximalwith respect to set inclusion are called τ -classes or just classes, when τ is understood. The family of all classes ofthe space (O, τ) is particularly interesting and is denoted by Hτ (O) . The family Hτ (O) is a covering of O [58].





https://en.wikipedia.org/wiki/Pseudometric_space

https://en.wikipedia.org/wiki/Mathematical_structure







41.10. TOLERANCE CLASSES AND PRECLASSES 163

The work on similarity by Poincaré and Zeeman presage the introduction of near sets[44][43] and research on similarityrelations, e.g.,[79]. In science and engineering, tolerance near sets are a practical application of the study of sets thatare near within some tolerance. A tolerance ε ∈ (0,∞] is directly related to the idea of closeness or resemblance(i.e., being within some tolerance) in comparing objects. By way of application of Poincaré's approach in definingvisual spaces and Zeeman’s approach to tolerance relations, the basic idea is to compare objects such as image patchesin the interior of digital images.

41.10.1 Examples

Simple ExampleThe following simple example demonstrates the construction of tolerance classes from real data. Consider the 20objects in the table below with |Φ| = 1 .

Let a tolerance relation be defined as

∼=ε= {(x, y) ∈ O ×O : ∥ Φ(x)− Φ(y) ∥2≤ ε}

Then, setting ε = 0.1 gives the following tolerance classes:

H∼=ε(O) ={{x1, x8, x10, x11}, {x1, x9, x10, x11, x14},{x2, x7, x18, x19},{x3, x12, x17},{x4, x13, x20}, {x4, x18},{x5, x6, x15, x16}, {x5, x6, x15, x20},{x6, x13, x20}}.

Observe that each object in a tolerance class satisfies the condition ∥ Φ(x)−Φ(y) ∥2≤ ε , and that almost all of theobjects appear in more than one class. Moreover, there would be twenty classes if the indiscernibility relation wasused since there are no two objects with matching descriptions.Image Processing Example

Figure 7. Example of images that are near each other. (a) and (b) Images from the freely available LeavesDataset (see, e.g.,www.vision.caltech.edu/archive.html).

The following example provides an example based on digital images. Let a subimage be defined as a small subset ofpixels belonging to a digital image such that the pixels contained in the subimage form a square. Then, let the sets Xand Y respectively represent the subimages obtained from two different images, and let O = {X ∪ Y } . Finally, letthe description of an object be given by the Green component in the RGB color model. The next step is to find all





https://en.wikipedia.org/wiki/Pixel

https://en.wikipedia.org/wiki/RGB_color_model


the tolerance classes using the tolerance relation defined in the previous example. Using this information, toleranceclasses can be formed containing objects that have similar (within some small ε ) values for the Green componentin the RGB colour model. Furthermore, images that are near (similar) to each other should have tolerance classesdivided among both images (instead of a tolerance classes contained solely in one of the images). For example, thefigure accompanying this example shows a subset of the tolerance classes obtained from two leaf images. In thisfigure, each tolerance class is assigned a separate colour. As can be seen, the two leaves share similar toleranceclasses. This example highlights a need to measure the degree of nearness of two sets.

41.11 Nearness measure

Let (U,RδΦ,ε) denote a particular descriptive pseudometric EF-proximal relator space equipped with the proximityrelation δΦ,ε and with nonempty subsets X,Y ∈ 2U and with the tolerance relation ∼=Φ,ε defined in terms of a set ofprobes Φ and with ε ∈ (0,∞] , where

Figure 8. Examples of degree of nearness between two sets: (a) High degree of nearness, and (b) Low degree of nearness.

≃Φ,ε= {(x, y) ∈ U × U | |Φ(x)− Φ(y)| ≤ ε}.

Further, assume Z = X ∪ Y and let HτΦ,ε(Z) denote the family of all classes in the space (Z,≃Φ,ε) .Let A ⊆ X,B ⊆ Y . The distance D

tNM: 2U × 2U :−→ [0,∞] is defined by

DtNM

(X,Y ) =

{1− tNM(A,B), if X and Y are not empty,∞, if X or Y is empty,

where

tNM(A,B) =

( ∑C∈HτΦ,ε

(Z)

|C|

)−1

·∑

C∈HτΦ,ε(Z)

|C|min(|C ∩A|, |[C ∩B|)max(|C ∩A|, |C ∩B|)

.

The details concerning tNM are given in[14][16][17]. The idea behind tNM is that sets that are similar should havea similar number of objects in each tolerance class. Thus, for each tolerance class obtained from the covering ofZ = X ∪Y , tNM counts the number of objects that belong toX and Y and takes the ratio (as a proper fraction) oftheir cardinalities. Furthermore, each ratio is weighted by the total size of the tolerance class (thus giving importanceto the larger classes) and the final result is normalized by dividing by the sum of all the cardinalities. The range oftNM is in the interval [0,1], where a value of 1 is obtained if the sets are equivalent (based on object descriptions)and a value of 0 is obtained if they have no descriptions in common.As an example of the degree of nearness between two sets, consider figure below in which each image consists of twosets of objects, X and Y . Each colour in the figures corresponds to a set where all the objects in the class share the




41.12. NEAR SET EVALUATION AND RECOGNITION (NEAR) SYSTEM 165

same description. The idea behind tNM is that the nearness of sets in a perceptual system is based on the cardinalityof tolerance classes that they share. Thus, the sets in left side of the figure are closer (more near) to each other interms of their descriptions than the sets in right side of the figure.

41.12 Near set evaluation and recognition (NEAR) system

Figure 9. NEAR system GUI.

The Near set Evaluation and Recognition (NEAR) system, is a system developed to demonstrate practical applicationsof near set theory to the problems of image segmentation evaluation and image correspondence. It was motivatedby a need for a freely available software tool that can provide results for research and to generate interest in nearset theory. The system implements a Multiple Document Interface (MDI) where each separate processing task isperformed in its own child frame. The objects (in the near set sense) in this system are subimages of the images beingprocessed and the probe functions (features) are image processing functions defined on the subimages. The systemwas written in C++ and was designed to facilitate the addition of new processing tasks and probe functions. Currently,the system performs six major tasks, namely, displaying equivalence and tolerance classes for an image, performingsegmentation evaluation, measuring the nearness of two images, performing Content Based Image Retrieval (CBIR),and displaying the output of processing an image using a specific probe function.

41.13 Proximity System

The Proximity System is an application developed to demonstrate descriptive-based topological approaches to near-ness and proximity within the context of digital image analysis. The Proximity System grew out of the work of S.Naimpally and J. Peters on Topological Spaces. The Proximity System was written in Java and is intended to run intwo different operating environments, namely on Android smartphones and tablets, as well as desktop platforms run-


Figure 10. The Proximity System.

ning the Java Virtual Machine. With respect to the desktop environment, the Proximity System is a cross-platformJava application for Windows, OSX, and Linux systems, which has been tested on Windows 7 and Debian Linuxusing the Sun Java 6 Runtime. In terms of the implementation of the theoretical approaches, both the Android andthe desktop based applications use the same back-end libraries to perform the description-based calculations, wherethe only differences are the user interface and the Android version has less available features due to restrictions onsystem resources.

41.14 See also• Alternative set theory

• Category:Mathematical relations

• Category:Topology

• Feature vector

• Proximity space

• Rough set

• Topology

41.15 Notes1. ^ J.R. Isbell observed that the notions near and far are important in a uniform space. Sets A,B are far

(uniformaly distal), provided the {A,B} is a discrete collection. A nonempty set U is a uniform neighbour-hood of a set A , provided the complement of U is far from U . See, §33 in [23]

2. ^ The intuition that led to the discovery of descriptively near sets is given in Pawlak, Z.;Peters, J.F. (2002,2007) “Jak blisko (How Near)". Systemy Wspomagania Decyzji I 57 (109)

3. ^ Descriptively near sets are introduced in[48]. The connections between traditional EF-proximity and descrip-tive EF-proximity are explored in [37].

https://en.wikipedia.org/wiki/Alternative_set_theory

https://en.wikipedia.org/wiki/Category:Mathematical_relations

https://en.wikipedia.org/wiki/Category:Topology

https://en.wikipedia.org/wiki/Feature_vector


https://en.wikipedia.org/wiki/Rough_set


https://en.wikipedia.org/wiki/Near_sets#ref_A_grp_2_1


https://en.wikipedia.org/wiki/Near_sets#ref_B_grp_2_1

https://en.wikipedia.org/wiki/Near_sets#ref_C_grp_2_1




4. ^ Reminiscent of M. Pavel’s approach, descriptions of members of sets objects are defined relative to vectorsof values obtained from real-valued functions called probes. See, Pavel, M. (1993). Fundamentals of patternrecognition. 2nd ed. New York: Marcel Dekker, for the introduction of probe functions considered in thecontext of image registration.

5. ^ A non-spatial view of near sets appears in, C.J. Mozzochi, M.S. Gagrat, and S.A. Naimpally, Symmetricgeneralized topological structures, Exposition Press, Hicksville, NY, 1976., and, more recently, nearness ofdisjoint sets X and Y based on resemblance between pairs of elements x ∈ X, y ∈ Y (i.e. x and y havesimilar feature vectors ϕ(x),ϕ(y) and the norm ∥ ϕ(x)− ϕ(y) ∥p< ε ) See, e.g.,[43][42][53].

6. ^ The basic facts about closure of a set were first pointed out by M. Fréchet in[11], and elaborated by B. Knasterand C. Kuratowski in[25].

7. ^Observe that up to the 1970s, proximity meant EF-proximity, since this is the one that was studied intensively.The pre-1970 work on proximity spaces is exemplified by the series of papers by J. M. Smirnov during thefirst half of the 1950s[68][67][69][70], culminating in the compendious collection of results by S.A. Naimpally andB.D. Warrack[34]. But in view of later developments, there is a need to distinguish between various proximities.A basic proximity or Čech-proximity was introduced by E. Čech during the late 1930s (see §25 A.1, pp. 439-440 in [78]). The conditions for the non-symmetric case for a proximity were introduced by S. Leader[28] andfor the symmetric case by M.W. Lodato[29][30][31].

41.16 References1. ^ Adámek, J.; Herrlich, H.; Strecker, G. E. (1990). Abstract and concrete categories. London: Wiley-

Interscience. pp. ix+482.

2. ^ Beer, G. (1993), “Topologies on closed and closed convex sets”, London, UK: Kluwer Academic Pub., pp.xi + 340pp. Missing or empty |title= (help)

3. ^ Bentley, H. L.; Colebunders, E.; Vandermissen, E. (2009), “A convenient setting for completions and func-tion spaces”, in Mynard, F.; Pearl, E., Contemporary Mathematics, Providence, RI: American MathematicalSociety, pp. 37–88 Missing or empty |title= (help)

4. ^ Cameron, P.; Hockingand, J. G.; Naimpally, S. A. (1974). “Nearness–a better approach to continuity andlimits”. American Mathematical Monthly 81 (7): 739–745. doi:10.2307/2319561.

5. ^ Di Concilio, A. (2008), “Action, uniformity and proximity”, in Naimpally, S. A.; Di Maio, G., Theory andApplications of Proximity, Nearness and Uniformity, Seconda Università di Napoli, Napoli: Prentice-Hall, pp.71–88 Missing or empty |title= (help)

6. ^ a b Di Concilio, A. (2009). “Proximity: A powerful tool in extension theory, function spaces, hyperspaces,boolean algebras and point-free geometry”. ContemporaryMathematics 486: 89–114. doi:10.1090/conm/486/09508.

7. ^ Devi, R.; Selvakumar, A.; Vigneshwaran, M. (2010). " (I, γ) -generalized semi-closed sets in topologicalspaces”. FILOMAT 24 (1): 97–100. doi:10.2298/fil1001097d.

8. ^ a b c Efremovič, V. A. (1952). “The geometry of proximity I (in Russian)". Mat. Sb. (N.S.) 31 (73): 189–200.

9. ^ Peters, J. F. (2008). “A note on a-open sets and e ∗ -sets”. FILOMAT 22 (1): 89–96.

10. ^ Fechner, G. T. (1966). Elements of Psychophysics, vol. I. London, UK: Hold, Rinehart & Winston. pp. H.E. Adler’s trans. of Elemente der Psychophysik, 1860.

11. ^ Fréchet, M. (1906). “Sur quelques points du calcul fonctionnel”. Rend. Circ. Mat. Palermo 22: 1–74.doi:10.1007/bf03018603.

12. ^ Grätzer, G.; Wenzel, G. H. (1989). “Tolerances, covering systems, and the axiom of choice”. ArchivumMathematicum 25 (1-2): 27–34.

13. ^ Gupta, S.; Patnaik, K. (2008). “Enhancing performance of face recognition systems by using near set ap-proach for selecting facial features”. Journal of Theoretical and Applied Information Technology 4 (5): 433–441.

https://en.wikipedia.org/wiki/Near_sets#ref_D_grp_2_1

https://en.wikipedia.org/wiki/Near_sets#ref_E_grp_2_1




https://en.wikipedia.org/wiki/Near_sets#ref_F_grp_2_1



https://en.wikipedia.org/wiki/Near_sets#ref_G_grp_2_1











https://en.wikipedia.org/wiki/Near_sets#ref_1_1


https://en.wikipedia.org/wiki/Help:CS1_errors#citation_missing_title





https://dx.doi.org/10.2307%252F2319561






https://dx.doi.org/10.1090%252Fconm%252F486%252F09508



https://dx.doi.org/10.2298%252Ffil1001097d









https://dx.doi.org/10.1007%252Fbf03018603




14. ^ a b Hassanien, A. E.; Abraham, A.; Peters, J. F.; Schaefer, G.; Henry, C. (2009). “Rough sets and near sets inmedical imaging: A review, IEEE”. Transactions on Information Technology in Biomedicine 13 (6): 955–968.doi:10.1109/TITB.2009.2017017.

15. ^ Hausdorff, F. (1914). Grundz¨uge der mengenlehre. Leipzig: Veit and Company. pp. viii + 476.

16. ^ Henry, C.; Peters, J. F. (2010). “Perception-based image classification, International”. Journal of IntelligentComputing and Cybernetics 3 (3): 410–430. doi:10.1108/17563781011066701.

17. ^ a b c d Henry, C. J. (2010), “Near sets: Theory and applications”, Ph.D. thesis, Dept. Elec. Comp. Eng., Uni.of MB, supervisor: J.F. Peters

18. ^ a b Henry, C.; Peters, J. F. (2011). “Arthritic hand-finger movement similarity measurements: Tolerance nearset approach”. Computational and Mathematical Methods in Medicine 2011: 1–14. doi:10.1155/2011/569898.

19. ^ Henry, C. J.; Ramanna, S. (2011). “Parallel Computation in Finding Near Neighbourhoods”. Lecture Notesin Computer Science,: 523–532.

20. ^ a b Herrlich, H. (1974). “A concept of nearness”. General Topology and its Applications 4: 191–212.doi:10.1016/0016-660x(74)90021-x.

21. ^ Hocking, J. G.; Naimpally, S. A. (2009), “Nearness—a better approach to continuity and limits”, AllahabadMathematical Society Lecture Note Series 3, Allahabad: The Allahabad Mathematical Society, pp. iv+66,ISBN 978-81-908159-1-8 Missing or empty |title= (help)

22. ^ Ïnan, E.; Öztürk, M. A. (2012). “Near groups on nearness approximation spaces”. Hacettepe Journal ofMathematics and Statistics 41 (4): 545–558.

23. ^ Isbell, J. R. (1964). Uniform spaces. Providence, Rhode Island: American Mathematical Society. pp. xi +175.

24. ^ Ivanova, V. M.; Ivanov, A. A. (1959). “Contiguity spaces and bicompact extensions of topological spaces(russian)". Dokl. Akad. Nauk SSSR 127: 20–22.

25. ^ Knaster, B.; Kuratowski, C. (1921). “Sur les ensembles connexes”. Fundamenta Mathematicae 2: 206–255.

26. ^Kovár, M. M. (2011). “A new causal topology and why the universe is co-compact”. arXive:1112.0817{[}math-ph]:1–15.

27. ^ Kuratowski, C. (1958), “Topologie i”, Warsaw: Panstwowe Wydawnictwo Naukowe, pp. XIII + 494pp.Missing or empty |title= (help)

28. ^ Leader, S. (1967). “Metrization of proximity spaces”. Proceedings of the American Mathematical Society18: 1084–1088. doi:10.2307/2035803.

29. ^ Lodato, M. W. (1962), “On topologically induced generalized proximity relations”, Ph.D. thesis, RutgersUniversity

30. ^ Lodato, M. W. (1964). “On topologically induced generalized proximity relations I”. Proceedings of theAmerican Mathematical Society 15: 417–422. doi:10.2307/2034517.

31. ^ Lodato, M. W. (1966). “On topologically induced generalized proximity relations II”. Pacific Journal ofMathematics 17: 131–135.

32. ^ MacLane, S. (1971). Categories for the working mathematician. Berlin: Springer. pp. v+262pp.

33. ^ Mozzochi, C. J.; Naimpally, S. A. (2009), “Uniformity and proximity”, Allahabad Mathematical SocietyLecture Note Series 2, Allahabad: The Allahabad Mathematical Society, pp. xii+153, ISBN 978-81-908159-1-8 Missing or empty |title= (help)

34. ^ Naimpally, S. A. (1970). Proximity spaces. Cambridge, UK: Cambridge University Press. pp. x+128. ISBN978-0-521-09183-1.

35. ^ Naimpally, S. A. (2009). Proximity approach to problems in topology and analysis. Munich, Germany:Oldenbourg Verlag. pp. ix + 204. ISBN 978-3-486-58917-7.




https://dx.doi.org/10.1109%252FTITB.2009.2017017




https://dx.doi.org/10.1108%252F17563781011066701








https://dx.doi.org/10.1155%252F2011%252F569898





https://dx.doi.org/10.1016%252F0016-660x%252874%252990021-x














https://dx.doi.org/10.2307%252F2035803




https://dx.doi.org/10.2307%252F2034517















36. ^ Naimpally, S. A.; Peters, J. F. (2013). “Preservation of continuity”. Scientiae Mathematicae Japonicae 76(2): 1–7.

37. ^ a b c d Naimpally, S. A.; Peters, J. F. (2013). Topology with Applications. Topological Spaces via Near andFar. Singapore: World Scientific.

38. ^ Naimpally, S. A.; Peters, J. F.; Wolski, M. (2013). Near set theory and applications. Special Issue inMathematics in Computer Science 7. Berlin: Springer. p. 136.

39. ^ Naimpally, S. A.; Warrack, B. D. (1970), “Proximity spaces”, Cambridge Tract in Mathematics 59, Cam-bridge, UK: Cambridge University Press, pp. x+128 Missing or empty |title= (help)

40. ^ Pal, S. K.; Peters, J. F. (2010). Rough fuzzy image analysis. Foundations and methodologies. London, UK,:CRC Press, Taylor & Francis Group. ISBN 9781439803295.

41. ^ Peters, J. F. (2009). “Tolerance near sets and image correspondence”. International Journal of Bio-InspiredComputation 1 (4): 239–245. doi:10.1504/ijbic.2009.024722.

42. ^ a b c Peters, J. F.; Wasilewski, P. (2009). “Foundations of near sets”. Information Sciences 179 (18): 3091–3109. doi:10.1016/j.ins.2009.04.018.

43. ^ a b c Peters, J. F. (2007). “Near sets. General theory about nearness of objects”. Applied MathematicalSciences 1 (53): 2609–2629.

44. ^ a b Peters, J. F. (2007). “Near sets. Special theory about nearness of objects”. Fundamenta Informaticae 75(1-4): 407–433.

45. ^ Peters, J. F. (2010). “Corrigenda and addenda: Tolerance near sets and image correspondence”. InternationalJournal of Bio-Inspired Computation 2 (5): 310–318. doi:10.1504/ijbic.2010.036157.

46. ^ Peters, J. F. (2011), “How near are Zdzisław Pawlak’s paintings? Merotopic distance between regions ofinterest”, in Skowron, A.; Suraj, S., Intelligent Systems Reference Library volume dedicated to Prof. ZdzisławPawlak, Berlin: Springer, pp. 1–19 Missing or empty |title= (help)

47. ^ Peters, J. F. (2011), “Sufficiently near sets of neighbourhoods”, in Yao, J. T.; Ramanna, S.; Wang, G. et al.,Lecture Notes in Artificial Intelligence 6954, Berlin: Springer, pp. 17–24 Missing or empty |title= (help);

48. ^ a b c d Peters, J. F. (2013). “Near sets: An introduction”. Mathematics in Computer Science 7 (1): 3–9.doi:10.1007/s11786-013-0149-6.

49. ^ a b c Peters, J. F. (2014). “Proximal relator spaces”. FILOMAT : 1–5 (in press).

50. ^ a b c d e Peters, J. F. (2014). Topology of Digital Images. Visual Pattern Discovery in Proximity Spaces 63.Springer. p. 342. ISBN 978-3-642-53844-5.

51. ^ a b Peters, J. F.; İnan, E.; Öztürk, M. A. (2014). “Spatial and descriptive isometries in proximity spaces”.General Mathematics Notes 21 (2): 125–134.

52. ^ Peters, J. F.; Naimpally, S. A. (2011). “Approach spaces for near families”. General Mathematics Notes 2(1): 159–164.

53. ^ a b c Peters, J. F.; Naimpally, S. A. (2011). General Mathematics Notes 2 (1): 159–164. Missing or empty|title= (help)

54. ^ Peters, J. F.; Puzio, L. (2009). “Image analysis with anisotropic wavelet-based nearness measures”. Interna-tional Journal of Computational Intelligence Systems 2 (3): 168–183. doi:10.1016/j.ins.2009.04.018.

55. ^ Peters, J. F.; Shahfar, S.; Ramanna, S.; Szturm, T. (2007), “Biologically-inspired adaptive learning: A nearset approach”, Frontiers in the Convergence of Bioscience and Information Technologies, Korea Missing orempty |title= (help)

56. ^ Peters, J. F.; Tiwari, S. (2011). “Approach merotopies and near filters. Theory and application”. GeneralMathematics Notes 3 (1): 32–45.

57. ^ Peters, J. F.; Tiwari, S. (2011). “Approach merotopies and near filters. Theory and application”. GeneralMathematics Notes 3 (1): 32–45.














https://dx.doi.org/10.1504%252Fijbic.2009.024722





https://dx.doi.org/10.1016%252Fj.ins.2009.04.018








https://dx.doi.org/10.1504%252Fijbic.2010.036157










https://dx.doi.org/10.1007%252Fs11786-013-0149-6


























58. ^ Peters, J. F.; Wasilewski, P. (2012). “Tolerance spaces: Origins, theoretical aspects and applications”. In-formation Sciences 195: 211–225. doi:10.1016/j.ins.2012.01.023.

59. ^ Picado, J. “Weil nearness spaces”. Portugaliae Mathematica 55 (2): 233–254.

60. ^ a b c Poincaré, J. H. (1895). “L'espace et la géomètrie”. Revue de m'etaphysique et de morale 3: 631–646.

61. ^ a b c Poincaré, J. H. (1902). “Sur certaines surfaces algébriques; troisième complément 'a l'analysis situs”.Bulletin de la Société de France 30: 49–70.

62. ^ a b Poincaré, J. H. (1913 & 2009). Dernières pensées, trans. by J.W. Bolduc as Mathematics and science: Lastessays. Paris & NY: Flammarion & Kessinger. Check date values in: |date= (help)

63. ^ a b Poincaré, J. H. (1894). “Sur la nature du raisonnement mathématique”. Revue de méaphysique et demorale 2: 371–384.

64. ^ a b Ramanna, S.; Meghdadi, A. H. (2009). “Measuring resemblances between swarm behaviours: A percep-tual tolerance near set approach”. Fundamenta Informaticae 95 (4): 533–552. doi:10.3233/FI-2009-163.

65. ^ a b Riesz, F. (1908). “Stetigkeitsbegriff und abstrakte mengenlehre”. Atti del IV Congresso Internazionale deiMatematici II: 182–109.

66. ^ Shreider, J. A. (1975). Equality, resemblance, and order. Russia: Mir Publishers. p. 279.

67. ^ a b c d Smirnov, J. M. (1952). “On proximity spaces”. Mat. Sb. (N.S.) 31 (73): 543–574 (English translation:Amer. Math. Soc. Trans. Ser. 2, 38, 1964, 5–35).

68. ^ Smirnov, J. M. (1952). “On proximity spaces in the sense of V.A. Efremovič". Math. Sb. (N.S.) 84:895–898, English translation: Amer. Math. Soc. Trans. Ser. 2, 38, 1964, 1–4.

69. ^ Smirnov, J. M. (1954). “On the completeness of proximity spaces. I.”. Trudy Moskov. Mat. Obšč 3:271–306, English translation: Amer. Math. Soc. Trans. Ser. 2, 38, 1964, 37–74.

70. ^ Smirnov, J. M. (1955). “On the completeness of proximity spaces. II.”. Trudy Moskov. Mat. Obšč 4:421–438, English translation: Amer. Math. Soc. Trans. Ser. 2, 38, 1964, 75–94.

71. ^ a b Sossinsky, A. B. (1986). “Tolerance space theory and some applications”. Acta Applicandae Mathemati-cae: An International Survey Journal on Applying Mathematics and Mathematical Applications 5 (2): 137–167.doi:10.1007/bf00046585.

72. ^ Száz, Á. (1997). “Uniformly, proximally and topologically compact relators”. Mathematica Pannonica 8 (1):103–116.

73. ^ Száz, Á. (1987). “Basic tools and mild continuities in relator spaces”. Acta Mathematica Hungarica 50:177–201. doi:10.1007/bf01903935.

74. ^ Száz, Á (2000). “An extension of Kelley’s closed relation theorem to relator spaces”. FILOMAT 14: 49–71.

75. ^ Tiwari, S. (2010), “Some aspects of general topology and applications. Approach merotopic structures andapplications”, Ph.D. thesis, Dept. of Math., Allahabad (U.P.), India, supervisor: M. khare

76. ^ a b Tiwari, S.; Peters, J. F. (2013). “A new approach to the study of extended metric spaces”. MathematicaAeterna 3 (7): 565–577.

77. ^ Tukey, J. W. (1940), “Convergence and uniformity in topology”, Annals of Mathematics Studies AM–2,Princeton, NJ: Princeton Univ. Press, p. 90 Missing or empty |title= (help)

78. ^ Čech, E. (1966). Topological spaces, revised ed. by Z. Frolik and M. Katětov. London: John Wiley & Sons.p. 893.

79. ^ Wasilewski, P. (2004), “On selected similarity relations and their applications into cognitive science”, Ph.D.thesis, Dept. Logic

80. ^ Wasilewski, P.; Peters, J. F.; Ramanna, S. (2011). “Perceptual tolerance intersection”. Transactions onRough Sets XIII: 159–174.













https://en.wikipedia.org/wiki/Help:CS1_errors#bad_date






https://dx.doi.org/10.3233%252FFI-2009-163






























81. ^ Weil, A. (1938), “Sur les espaces à structure uniforme et sur la topologie générale”, Actualités scientifiqueet industrielles, Paris: Harmann & cie Missing or empty |title= (help)

82. ^ Wolski, M. (2010). “Perception and classification. A note on near sets and rough sets”. Fundamenta Infor-maticae 101: 143–155.

83. ^ a b Zeeman, E. C. (1962), “The topology of the brain and visual perception”, in Fort, Jr., M. K., Topologyof 3-Manifolds and Related Topics, University of Georgia Institute Conference Proceedings (1962): Prentice-Hall, pp. 240–256 Missing or empty |title= (help)

41.17 Further reading• Naimpally, S. A.; Peters, J. F. (2013). Topology with Applications. Topological Spaces via Near and Far.

World Scientific Publishing . Co. Pte. Ltd. ISBN 978-981-4407-65-6.

• Naimpally, S. A.; Peters, J. F.; Wolski, M. (2013), "Near Set Theory and Applications", Mathematics inComputer Science 7 (1), Berlin: Springer Missing or empty |title= (help)

• Peters, J. F. (2014), "Topology of Digital Images. Visual Pattern Discovery in Proximity Spaces", IntelligentSystems Reference Library 63, Berlin: Springer Missing or empty |title= (help)

• Henry, C. J.; Peters, J. F. (2012), "Near set evaluation and recognition (NEAR) system V3.0", UM CI Labo-ratory Technical Report No. TR-2009-015, Computational Intelligence Laboratory, University of ManitobaMissing or empty |title= (help)

• Concilio, A. Di (2014). “Proximity: A powerful tool in extension theory, function spaces, hyperspaces, booleanalgebras and point-free geometry”. Computational Intelligence Laboratory, University of Manitoba. UM CILaboratory Technical Report No. TR-2009-021.

• Peters, J. F.; Naimpally, S. A. (2012). “Applications of near sets” (PDF). Notices of the AmericanMathematicalSociety 59 (4): 536–542. CiteSeerX: 10 .1 .1 .371 .7903.








http://www.ams.org/mathscinet/search/publdoc.html?pg1=INDI&s1=334268&vfpref=html&r=12&mx-pid=3075111



http://link.springer.com/journal/11786/7/1


http://link.springer.com/book/10.1007%5C%252F978-3-642-53845-2


http://wren.ee.umanitoba.ca/index.php?option=com_content&view=article&id=51&Itemid=78


http://www.ams.org/notices/201204/rtx120400536p.pdf

https://en.wikipedia.org/wiki/CiteSeer#CiteSeerX

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.371.7903

Chapter 42

Partial equivalence relation

In mathematics, a partial equivalence relation (often abbreviated as PER, in older literature also called restrictedequivalence relation) R on a set X is a relation that is symmetric and transitive. In other words, it holds for alla, b, c ∈ X that:

1. if aRb , then bRa (symmetry)

2. if aRb and bRc , then aRc (transitivity)

If R is also reflexive, then R is an equivalence relation.

42.1 Properties and applications

In a set-theoretic context, there is a simple structure to the general PER R on X : it is an equivalence relation on thesubset Y = {x ∈ X|xRx} ⊆ X . ( Y is the subset of X such that in the complement of Y ( X \ Y ) no element isrelated by R to any other.) By construction, R is reflexive on Y and therefore an equivalence relation on Y . Noticethat R is actually only true on elements of Y : if xRy , then yRx by symmetry, so xRx and yRy by transitivity.Conversely, given a subset Y of X, any equivalence relation on Y is automatically a PER on X.PERs are therefore used mainly in computer science, type theory and constructive mathematics, particularly to definesetoids, sometimes called partial setoids. The action of forming one from a type and a PER is analogous to theoperations of subset and quotient in classical set-theoretic mathematics.Every partial equivalence relation is a difunctional relation, but the converse does not hold.The algebraic notion of congruence can also be generalized to partial equivalences, yielding the notion of subcongruence,i.e. a homomorphic relation that is symmetric and transitive, but not necessarily reflexive.[1]

42.2 Examples

A simple example of a PER that is not an equivalence relation is the empty relation R = ∅ (unless X = ∅ , in whichcase the empty relation is an equivalence relation (and is the only relation on X )).

42.2.1 Euclidean parallelism

In the Euclidean plane, two lines m and n are parallel lines when m ∩ n = ∅. The symmetry of this relation is obviousand the transitivity can be proven in the Euclidean plane, thus Euclidean parallelism is a partial equivalence relation.Nevertheless, mathematicians developing affine geometry prefer the facility of an equivalence relation and thereforesometimes revise the definition of parallelism to allow a line to be parallel to itself, making the new relation of “affineparallelism” that is a reflexive relation.

172






https://en.wikipedia.org/wiki/Set-theoretic



https://en.wikipedia.org/wiki/Type_theory

https://en.wikipedia.org/wiki/Constructive_mathematics

https://en.wikipedia.org/wiki/Setoid

https://en.wikipedia.org/wiki/Difunctional


https://en.wikipedia.org/wiki/Subcongruence

https://en.wikipedia.org/wiki/Homomorphic_relation


https://en.wikipedia.org/wiki/Parallel_lines

https://en.wikipedia.org/wiki/Affine_geometry




42.2.2 Kernels of partial functions

For another example of a PER, consider a set A and a partial function f that is defined on some elements of A butnot all. Then the relation ≈ defined by

x ≈ y if and only if f is defined at x , f is defined at y , and f(x) = f(y)

is a partial equivalence relation but not an equivalence relation. It possesses the symmetry and transitivity properties,but it is not reflexive since if f(x) is not defined then x ̸≈ x — in fact, for such an x there is no y ∈ A such thatx ≈ y . (It follows immediately that the subset of A for which ≈ is an equivalence relation is precisely the subset onwhich f is defined.)

42.2.3 Functions respecting equivalence relations

Let X and Y be sets equipped with equivalence relations (or PERs) ≈X ,≈Y . For f, g : X → Y , define f ≈ g tomean:

∀x0 x1, x0 ≈X x1 ⇒ f(x0) ≈Y g(x1)

then f ≈ f means that f induces a well-defined function of the quotients X/ ≈X → Y / ≈Y . Thus, the PER≈ captures both the idea of definedness on the quotients and of two functions inducing the same function on thequotient.

42.3 References[1] J. Lambek (1996). “The Butterfly and the Serpent”. In Aldo Ursini, Paulo Agliano. Logic and Algebra. CRC Press. pp.

161–180. ISBN 978-0-8247-9606-8.

• Mitchell, John C. Foundations of programming languages. MIT Press, 1996.

• D.S. Scott. “Data types as lattices”. SIAM Journ. Comput., 3:523-587, 1976.

42.4 See also• Equivalence relation

• Binary relation




http://portal.acm.org/citation.cfm?id=237842



Chapter 43

Partial function

Not to be confused with partial function of a multilinear map or the mathematical concept of a piecewise function.

In mathematics, a partial function from X to Y (written as f: X ↛ Y) is a function f: X′ → Y, for some subset X′of X. It generalizes the concept of a function f: X → Y by not forcing f to map every element of X to an elementof Y (only some subset X′ of X). If X′ = X, then f is called a total function and is equivalent to a function. Partialfunctions are often used when the exact domain, X′, is not known (e.g. many functions in computability theory).Specifically, we will say that for any x ∈ X, either:

• f(x) = y ∈ Y (it is defined as a single element in Y) or

• f(x) is undefined.

For example we can consider the square root function restricted to the integers

g : Z → Z

g(n) =√n.

Thus g(n) is only defined for n that are perfect squares (i.e., 0, 1, 4, 9, 16, ...). So, g(25) = 5, but g(26) is undefined.

43.1 Basic concepts

There are two distinct meanings in current mathematical usage for the notion of the domain of a partial function.Most mathematicians, including recursion theorists, use the term “domain of f" for the set of all values x such thatf(x) is defined (X' above). But some, particularly category theorists, consider the domain of a partial function f:X →Y to be X, and refer to X' as the domain of definition. Similarly, the term range can refer to either the codomain orthe image of a function.Occasionally, a partial function with domain X and codomain Y is written as f: X ⇸ Y, using an arrow with verticalstroke.A partial function is said to be injective or surjective when the total function given by the restriction of the partialfunction to its domain of definition is. A partial function may be both injective and surjective.Because a function is trivially surjective when restricted to its image, the term partial bijection denotes a partialfunction which is injective.[1]

An injective partial function may be inverted to an injective partial function, and a partial function which is bothinjective and surjective has an injective function as inverse. Furthermore, a total function which is injective may beinverted to an injective partial function.The notion of transformation can be generalized to partial functions as well. A partial transformation is a functionf: A → B, where both A and B are subsets of some set X.[2]

174


https://en.wikipedia.org/wiki/Piecewise_function





https://en.wikipedia.org/wiki/Computability_theory



https://en.wikipedia.org/wiki/Square_number


https://en.wikipedia.org/wiki/Recursion_theory






https://en.wikipedia.org/wiki/Partial_bijection




43.2. TOTAL FUNCTION 175

43.2 Total function

Total function is a synonym for function. The use of the prefix “total” is to suggest that it is a special case over a largerset X of a partial function over a subset of X. For example, when considering the operation of morphism compositionin Concrete Categories, the composition operation ◦ : Hom(C) × Hom(C) → Hom(C) is a total function if andonly if Ob(C) has one element. The reason for this is that two morphisms f : X → Y and g : U → V can only becomposed as g ◦ f if Y = U , that is, the codomain of f must equal the domain of g .

43.3 Discussion and examples

The first diagram above represents a partial function that is not a total function since the element 1 in the left-hand setis not associated with anything in the right-hand set. Whereas, the second diagram represents a total function sinceevery element on the left-hand set is associated with exactly one element in the right hand set.

43.3.1 Natural logarithm

Consider the natural logarithm function mapping the real numbers to themselves. The logarithm of a non-positivereal is not a real number, so the natural logarithm function doesn't associate any real number in the codomain withany non-positive real number in the domain. Therefore, the natural logarithm function is not a total function whenviewed as a function from the reals to themselves, but it is a partial function. If the domain is restricted to only includethe positive reals (that is, if the natural logarithm function is viewed as a function from the positive reals to the reals),then the natural logarithm is a total function.

43.3.2 Subtraction of natural numbers

Subtraction of natural numbers (non-negative integers) can be viewed as a partial function:

f : N× N → N

f(x, y) = x− y.

It is defined only when x ≥ y .

43.3.3 Bottom element

In denotational semantics a partial function is considered as returning the bottom element when it is undefined.In computer science a partial function corresponds to a subroutine that raises an exception or loops forever. The IEEEfloating point standard defines a not-a-number value which is returned when a floating point operation is undefinedand exceptions are suppressed, e.g. when the square root of a negative number is requested.In a programming language where function parameters are statically typed, a function may be defined as a partialfunction because the language’s type system cannot express the exact domain of the function, so the programmerinstead gives it the smallest domain which is expressible as a type and contains the true domain.

43.3.4 In category theory

The category of sets and partial functions is equivalent to but not isomorphic with the category of pointed sets andpoint-preserving maps.[3] One textbook notes that “This formal completion of sets and partial maps by adding “im-proper,” “infinite” elements was reinvented many times, in particular, in topology (one-point compactification) andin theoretical computer science.”[4]

The category of sets and partial bijections is equivalent to its dual.[5] It is the prototypical inverse category.[6]






https://en.wikipedia.org/wiki/Positive_reals



https://en.wikipedia.org/wiki/Denotational_semantics

https://en.wikipedia.org/wiki/Bottom_element


https://en.wikipedia.org/wiki/IEEE_floating_point

https://en.wikipedia.org/wiki/IEEE_floating_point

https://en.wikipedia.org/wiki/Not-a-number

https://en.wikipedia.org/wiki/Programming_language

https://en.wikipedia.org/wiki/Statically_typed

https://en.wikipedia.org/wiki/Type_system

https://en.wikipedia.org/wiki/Equivalence_of_categories

https://en.wikipedia.org/wiki/Isomorphism_of_categories

https://en.wikipedia.org/wiki/Pointed_set

https://en.wikipedia.org/wiki/One-point_compactification

https://en.wikipedia.org/wiki/Theoretical_computer_science

https://en.wikipedia.org/wiki/Opposite_category

https://en.wikipedia.org/wiki/Inverse_category

176 CHAPTER 43. PARTIAL FUNCTION

43.3.5 In abstract algebra

Partial algebra generalizes the notion of universal algebra to partial operations. An example would be a field, in whichthe multiplicative inversion is the only proper partial operation (because division by zero is not defined).[7]

The set of all partial functions (partial transformations) on a given base X set forms a regular semigroup called thesemigroup of all partial transformations (or the partial transformation semigroup on X), typically denoted by PT X

.[8][9][10] The set of all partial bijections on X forms the symmetric inverse semigroup.[8][9]

43.4 See also• Bijection

• Injective function


• Multivalued function

• Densely defined operator

43.5 References[1] Christopher Hollings (2014). Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups.

American Mathematical Society. p. 251. ISBN 978-1-4704-1493-1.

[2] Christopher Hollings (2014). Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups.American Mathematical Society. p. 251. ISBN 978-1-4704-1493-1.

[3] Lutz Schröder (2001). “Categories: a free tour”. In Jürgen Koslowski and Austin Melton. Categorical Perspectives. SpringerScience & Business Media. p. 10. ISBN 978-0-8176-4186-3.

[4] Neal Koblitz; B. Zilber; Yu. I. Manin (2009). A Course in Mathematical Logic for Mathematicians. Springer Science &Business Media. p. 290. ISBN 978-1-4419-0615-1.

[5] Francis Borceux (1994). Handbook of Categorical Algebra: Volume 2, Categories and Structures. Cambridge UniversityPress. p. 289. ISBN 978-0-521-44179-7.

[6] Marco Grandis (2012). Homological Algebra: The Interplay of Homology with Distributive Lattices and Orthodox Semi-groups. World Scientific. p. 55. ISBN 978-981-4407-06-9.

[7] Peter Burmeister (1993). “Partial algebras – an introductory survey”. In Ivo G. Rosenberg and Gert Sabidussi. Algebrasand Orders. Springer Science & Business Media. ISBN 978-0-7923-2143-9.

[8] Alfred Hoblitzelle Clifford; G. B. Preston (1967). The Algebraic Theory of Semigroups. Volume II. American MathematicalSoc. p. xii. ISBN 978-0-8218-0272-4.

[9] Peter M. Higgins (1992). Techniques of semigroup theory. Oxford University Press, Incorporated. p. 4. ISBN 978-0-19-853577-5.

[10] Olexandr Ganyushkin; Volodymyr Mazorchuk (2008). Classical Finite Transformation Semigroups: An Introduction.Springer Science & Business Media. pp. 16 and 24. ISBN 978-1-84800-281-4.

• Martin Davis (1958), Computability and Unsolvability, McGraw–Hill Book Company, Inc, New York. Re-published by Dover in 1982. ISBN 0-486-61471-9.

• Stephen Kleene (1952), Introduction to Meta-Mathematics, North-Holland Publishing Company, Amsterdam,Netherlands, 10th printing with corrections added on 7th printing (1974). ISBN 0-7204-2103-9.

• Harold S. Stone (1972), Introduction to Computer Organization and Data Structures, McGraw–Hill Book Com-pany, New York.

https://en.wikipedia.org/wiki/Partial_algebra




https://en.wikipedia.org/wiki/Division_by_zero


https://en.wikipedia.org/wiki/Regular_semigroup

https://en.wikipedia.org/wiki/Symmetric_inverse_semigroup





https://en.wikipedia.org/wiki/Densely_defined_operator











http://books.google.com/books?id=5i2v9q0m5XAC&pg=PA289



http://books.google.com/books?id=TWqhelao4KsC&pg=PA55

http://books.google.com/books?id=TWqhelao4KsC&pg=PA55





http://books.google.com/books?id=756KAwAAQBAJ&pg=PR12








https://en.wikipedia.org/wiki/Martin_Davis


https://en.wikipedia.org/wiki/Stephen_Kleene


https://en.wikipedia.org/wiki/Harold_S._Stone

Chapter 44

Partially ordered set

{x,y,z}

{y,z}{x,z}{x,y}

{y} {z}{x}

Ø

The Hasse diagram of the set of all subsets of a three-element set {x, y, z}, ordered by inclusion. Sets on the same horizontal leveldon't share a precedence relationship. Other pairs, such as {x} and {y,z}, do not either.

In mathematics, especially order theory, a partially ordered set (or poset) formalizes and generalizes the intuitiveconcept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together witha binary relation that indicates that, for certain pairs of elements in the set, one of the elements precedes the other.Such a relation is called a partial order to reflect the fact that not every pair of elements need be related: for somepairs, it may be that neither element precedes the other in the poset. Thus, partial orders generalize the more familiartotal orders, in which every pair is related. A finite poset can be visualized through its Hasse diagram, which depictsthe ordering relation.[1]

A familiar real-life example of a partially ordered set is a collection of people ordered by genealogical descendancy.Some pairs of people bear the descendant-ancestor relationship, but other pairs bear no such relationship.

177









https://en.wikipedia.org/wiki/Genealogy

178 CHAPTER 44. PARTIALLY ORDERED SET


A (non-strict) partial order[2] is a binary relation "≤" over a set P which is reflexive, antisymmetric, and transitive,i.e., which satisfies for all a, b, and c in P:

• a ≤ a (reflexivity);

• if a ≤ b and b ≤ a then a = b (antisymmetry);

• if a ≤ b and b ≤ c then a ≤ c (transitivity).

In other words, a partial order is an antisymmetric preorder.A set with a partial order is called a partially ordered set (also called a poset). The term ordered set is sometimesalso used, as long as it is clear from the context that no other kind of order is meant. In particular, totally ordered setscan also be referred to as “ordered sets”, especially in areas where these structures are more common than posets.For a, b, elements of a partially ordered set P, if a ≤ b or b ≤ a, then a and b are comparable. Otherwise they areincomparable. In the figure on top-right, e.g. {x} and {x,y,z} are comparable, while {x} and {y} are not. A partialorder under which every pair of elements is comparable is called a total order or linear order; a totally orderedset is also called a chain (e.g., the natural numbers with their standard order). A subset of a poset in which no twodistinct elements are comparable is called an antichain (e.g. the set of singletons {{x}, {y}, {z}} in the top-rightfigure). An element a is said to be covered by another element b, written a<:b, if a is strictly less than b and no thirdelement c fits between them; formally: if both a≤b and a≠b are true, and a≤c≤b is false for each c with a≠c≠b. Amore concise definition will be given below using the strict order corresponding to "≤". For example, {x} is coveredby {x,z} in the top-right figure, but not by {x,y,z}.

44.2 Examples

Standard examples of posets arising in mathematics include:

• The real numbers ordered by the standard less-than-or-equal relation ≤ (a totally ordered set as well).

• The set of subsets of a given set (its power set) ordered by inclusion (see the figure on top-right). Similarly, theset of sequences ordered by subsequence, and the set of strings ordered by substring.

• The set of natural numbers equipped with the relation of divisibility.

• The vertex set of a directed acyclic graph ordered by reachability.

• The set of subspaces of a vector space ordered by inclusion.

• For a partially ordered set P, the sequence space containing all sequences of elements from P, where sequencea precedes sequence b if every item in a precedes the corresponding item in b. Formally, (an)n∈ℕ ≤ (bn) ∈ℕif and only if a ≤ b for all n in ℕ, i.e. a componentwise order.

• For a set X and a partially ordered set P, the function space containing all functions from X to P, where f ≤ gif and only if f(x) ≤ g(x) for all x in X.

• A fence, a partially ordered set defined by an alternating sequence of order relations a < b > c < d ...

44.3 Extrema

There are several notions of “greatest” and “least” element in a poset P, notably:













https://en.wikipedia.org/wiki/Antichain


https://en.wikipedia.org/wiki/Covering_relation







https://en.wikipedia.org/wiki/Subsequence

https://en.wikipedia.org/wiki/String_(computer_science)

https://en.wikipedia.org/wiki/Substring


https://en.wikipedia.org/wiki/Divisor#Divisibility_of_numbers

https://en.wikipedia.org/wiki/Directed_acyclic_graph

https://en.wikipedia.org/wiki/Reachability

https://en.wikipedia.org/wiki/Linear_subspace


https://en.wikipedia.org/wiki/Sequence_space


https://en.wikipedia.org/wiki/Componentwise_order

https://en.wikipedia.org/wiki/Function_space

https://en.wikipedia.org/wiki/Fence_(mathematics)

44.4. ORDERS ON THE CARTESIAN PRODUCT OF PARTIALLY ORDERED SETS 179

• Greatest element and least element: An element g in P is a greatest element if for every element a in P, a ≤ g.An element m in P is a least element if for every element a in P, a ≥ m. A poset can only have one greatest orleast element.

• Maximal elements and minimal elements: An element g in P is a maximal element if there is no element a inP such that a > g. Similarly, an element m in P is a minimal element if there is no element a in P such that a <m. If a poset has a greatest element, it must be the unique maximal element, but otherwise there can be morethan one maximal element, and similarly for least elements and minimal elements.

• Upper and lower bounds: For a subset A of P, an element x in P is an upper bound of A if a ≤ x, for eachelement a in A. In particular, x need not be in A to be an upper bound of A. Similarly, an element x in P is alower bound of A if a ≥ x, for each element a in A. A greatest element of P is an upper bound of P itself, anda least element is a lower bound of P.

For example, consider the positive integers, ordered by divisibility: 1 is a least element, as it divides all other elements;on the other hand this poset does not have a greatest element (although if one would include 0 in the poset, whichis a multiple of any integer, that would be a greatest element; see figure). This partially ordered set does not evenhave any maximal elements, since any g divides for instance 2g, which is distinct from it, so g is not maximal. If thenumber 1 is excluded, while keeping divisibility as ordering on the elements greater than 1, then the resulting posetdoes not have a least element, but any prime number is a minimal element for it. In this poset, 60 is an upper bound(though not a least upper bound) of the subset {2,3,5,10}, which does not have any lower bound (since 1 is not in theposet); on the other hand 2 is a lower bound of the subset of powers of 2, which does not have any upper bound.

44.4 Orders on the Cartesian product of partially ordered sets

In order of increasing strength, i.e., decreasing sets of pairs, three of the possible partial orders on the Cartesianproduct of two partially ordered sets are (see figures):

• the lexicographical order: (a,b) ≤ (c,d) if a < c or (a = c and b ≤ d);

• the product order: (a,b) ≤ (c,d) if a ≤ c and b ≤ d;

• the reflexive closure of the direct product of the corresponding strict orders: (a,b) ≤ (c,d) if (a < c and b < d)or (a = c and b = d).

All three can similarly be defined for the Cartesian product of more than two sets.Applied to ordered vector spaces over the same field, the result is in each case also an ordered vector space.See also orders on the Cartesian product of totally ordered sets.

44.5 Sums of partially ordered sets

Another way to combine two posets is the ordinal sum[3] (or linear sum[4]), Z = X ⊕ Y, defined on the union of theunderlying sets X and Y by the order a ≤Z b if and only if:

• a, b ∈ X with a ≤X b, or

• a, b ∈ Y with a ≤Y b, or

• a ∈ X and b ∈ Y.

If two posets are well-ordered, then so is their ordinal sum.[5]

https://en.wikipedia.org/wiki/Greatest_element


https://en.wikipedia.org/wiki/Upper_and_lower_bounds





https://en.wikipedia.org/wiki/Lexicographical_order


https://en.wikipedia.org/wiki/Reflexive_closure

https://en.wikipedia.org/wiki/Direct_product#Direct_product_of_binary_relations

https://en.wikipedia.org/wiki/Ordered_vector_space


https://en.wikipedia.org/wiki/Total_order#Orders_on_the_Cartesian_product_of_totally_ordered_sets

https://en.wikipedia.org/wiki/Well-ordered


44.6 Strict and non-strict partial orders

In some contexts, the partial order defined above is called a non-strict (or reflexive, or weak) partial order. In thesecontexts a strict (or irreflexive) partial order "<" is a binary relation that is irreflexive, transitive and asymmetric,i.e. which satisfies for all a, b, and c in P:

• not a < a (irreflexivity),

• if a < b and b < c then a < c (transitivity), and

• if a < b then not b < a (asymmetry; implied by irreflexivity and transitivity[6]).

There is a 1-to-1 correspondence between all non-strict and strict partial orders.If "≤" is a non-strict partial order, then the corresponding strict partial order "<" is the irreflexive kernel given by:

a < b if a ≤ b and a ≠ b

Conversely, if "<" is a strict partial order, then the corresponding non-strict partial order "≤" is the reflexive closuregiven by:

a ≤ b if a < b or a = b.

This is the reason for using the notation "≤".Using the strict order "<", the relation "a is covered by b" can be equivalently rephrased as "a<b, but not a<c<b forany c". Strict partial orders are useful because they correspond more directly to directed acyclic graphs (dags): everystrict partial order is a dag, and the transitive closure of a dag is both a strict partial order and also a dag itself.

44.7 Inverse and order dual

The inverse or converse ≥ of a partial order relation ≤ satisfies x≥y if and only if y≤x. The inverse of a partialorder relation is reflexive, transitive, and antisymmetric, and hence itself a partial order relation. The order dual of apartially ordered set is the same set with the partial order relation replaced by its inverse. The irreflexive relation > isto ≥ as < is to ≤.Any one of the four relations ≤, <, ≥, and > on a given set uniquely determines the other three.In general two elements x and y of a partial order may stand in any of four mutually exclusive relationships to eachother: either x < y, or x = y, or x > y, or x and y are incomparable (none of the other three). A totally ordered set is onethat rules out this fourth possibility: all pairs of elements are comparable and we then say that trichotomy holds. Thenatural numbers, the integers, the rationals, and the reals are all totally ordered by their algebraic (signed) magnitudewhereas the complex numbers are not. This is not to say that the complex numbers cannot be totally ordered; wecould for example order them lexicographically via x+iy < u+iv if and only if x < u or (x = u and y < v), but this is notordering by magnitude in any reasonable sense as it makes 1 greater than 100i. Ordering them by absolute magnitudeyields a preorder in which all pairs are comparable, but this is not a partial order since 1 and i have the same absolutemagnitude but are not equal, violating antisymmetry.

44.8 Mappings between partially ordered sets

Given two partially ordered sets (S,≤) and (T,≤), a function f: S → T is called order-preserving, or monotone,or isotone, if for all x and y in S, x≤y implies f(x) ≤ f(y). If (U,≤) is also a partially ordered set, and both f: S→ T and g: T → U are order-preserving, their composition (g∘f): S → U is order-preserving, too. A function f:S → T is called order-reflecting if for all x and y in S, f(x) ≤ f(y) implies x≤y. If f is both order-preserving andorder-reflecting, then it is called an order-embedding of (S,≤) into (T,≤). In the latter case, f is necessarily injective,since f(x) = f(y) implies x ≤ y and y ≤ x. If an order-embedding between two posets S and T exists, one says that Scan be embedded into T. If an order-embedding f: S → T is bijective, it is called an order isomorphism, and the




https://en.wikipedia.org/wiki/Irreflexive_kernel













https://en.wikipedia.org/wiki/Order-preserving

https://en.wikipedia.org/wiki/Monotonic_function#Monotonicity_in_order_theory


https://en.wikipedia.org/wiki/Order-embedding



https://en.wikipedia.org/wiki/Order_isomorphism

44.9. NUMBER OF PARTIAL ORDERS 181

partial orders (S,≤) and (T,≤) are said to be isomorphic. Isomorphic orders have structurally similar Hasse diagrams(cf. right picture). It can be shown that if order-preserving maps f: S → T and g: T → S exist such that g∘f and f∘gyields the identity function on S and T, respectively, then S and T are order-isomorphic. [7]

For example, a mapping f: ℕ → ℙ(ℕ) from the set of natural numbers (ordered by divisibility) to the power set ofnatural numbers (ordered by set inclusion) can be defined by taking each number to the set of its prime divisors. Itis order-preserving: if x divides y, then each prime divisor of x is also a prime divisor of y. However, it is neitherinjective (since it maps both 12 and 6 to {2,3}) nor order-reflecting (since besides 12 doesn't divide 6). Takinginstead each number to the set of its prime power divisors defines a map g: ℕ → ℙ(ℕ) that is order-preserving, order-reflecting, and hence an order-embedding. It is not an order-isomorphism (since it e.g. doesn't map any number tothe set {4}), but it can be made one by restricting its codomain to g(ℕ). The right picture shows a subset of ℕ and itsisomorphic image under g. The construction of such an order-isomorphism into a power set can be generalized to awide class of partial orders, called distributive lattices, see "Birkhoff’s representation theorem".

44.9 Number of partial orders

Partially ordered set of set of all subsets of a six-element set {a, b, c, d, e, f}, ordered by the subset relation.

Sequence A001035 in OEIS gives the number of partial orders on a set of n labeled elements:The number of strict partial orders is the same as that of partial orders.If we count only up to isomorphism, we get 1, 1, 2, 5, 16, 63, 318, … (sequence A000112 in OEIS).

44.10 Linear extension

A partial order ≤* on a set X is an extension of another partial order ≤ on X provided that for all elements x and yof X, whenever x ≤ y, it is also the case that x ≤* y. A linear extension is an extension that is also a linear (i.e., total)order. Every partial order can be extended to a total order (order-extension principle).[8]

In computer science, algorithms for finding linear extensions of partial orders (represented as the reachability ordersof directed acyclic graphs) are called topological sorting.




https://en.wikipedia.org/wiki/Prime_divisor

https://en.wikipedia.org/wiki/Prime_power

https://en.wikipedia.org/wiki/Injective_function#Injections_may_be_made_invertible

https://en.wikipedia.org/wiki/Distributive_lattice

https://en.wikipedia.org/wiki/Birkhoff%2527s_representation_theorem







https://en.wikipedia.org/wiki/Linear_extension

https://en.wikipedia.org/wiki/Order-extension_principle




https://en.wikipedia.org/wiki/Topological_sorting


44.11 In category theory

Every poset (and every preorder) may be considered as a category in which every hom-set has at most one element.More explicitly, let hom(x, y) = {(x, y)} if x ≤ y (and otherwise the empty set) and (y, z)∘(x, y) = (x, z). Posets areequivalent to one another if and only if they are isomorphic. In a poset, the smallest element, if it exists, is an initialobject, and the largest element, if it exists, is a terminal object. Also, every preordered set is equivalent to a poset.Finally, every subcategory of a poset is isomorphism-closed.

44.12 Partial orders in topological spaces

Main article: Partially ordered space

If P is a partially ordered set that has also been given the structure of a topological space, then it is customary toassume that {(a, b) : a ≤ b} is a closed subset of the topological product space P ×P . Under this assumption partialorder relations are well behaved at limits in the sense that if ai → a , bi → b and ai ≤ bi for all i, then a ≤ b.[9]

44.13 Interval

For a ≤ b, the closed interval [a,b] is the set of elements x satisfying a ≤ x ≤ b (i.e. a ≤ x and x ≤ b). It contains atleast the elements a and b.Using the corresponding strict relation "<", the open interval (a,b) is the set of elements x satisfying a < x < b (i.e. a< x and x < b). An open interval may be empty even if a < b. For example, the open interval (1,2) on the integers isempty since there are no integers i such that 1 < i < 2.Sometimes the definitions are extended to allow a > b, in which case the interval is empty.The half-open intervals [a,b) and (a,b] are defined similarly.A poset is locally finite if every interval is finite. For example, the integers are locally finite under their natural order-ing. The lexicographical order on the cartesian product ℕ×ℕ is not locally finite, since e.g. (1,2)≤(1,3)≤(1,4)≤(1,5)≤...≤(2,1).Using the interval notation, the property "a is covered by b" can be rephrased equivalently as [a,b] = {a,b}.This concept of an interval in a partial order should not be confused with the particular class of partial orders knownas the interval orders.

44.14 See also

• antimatroid, a formalization of orderings on a set that allows more general families of orderings than posets

• causal set

• comparability graph

• complete partial order

• directed set

• graded poset

• incidence algebra

• lattice

• locally finite poset

• Möbius function on posets

• ordered group



https://en.wikipedia.org/wiki/Equivalence_of_categories

https://en.wikipedia.org/wiki/Isomorphic

https://en.wikipedia.org/wiki/Initial_object

https://en.wikipedia.org/wiki/Initial_object

https://en.wikipedia.org/wiki/Terminal_object

https://en.wikipedia.org/wiki/Isomorphism-closed

https://en.wikipedia.org/wiki/Partially_ordered_space


https://en.wikipedia.org/wiki/Closed_(mathematics)

https://en.wikipedia.org/wiki/Product_space

https://en.wikipedia.org/wiki/Limit_of_a_sequence

https://en.wikipedia.org/wiki/Interval_(mathematics)

https://en.wikipedia.org/wiki/Open_interval

https://en.wikipedia.org/wiki/Locally_finite_poset


https://en.wikipedia.org/wiki/Interval_order

https://en.wikipedia.org/wiki/Antimatroid

https://en.wikipedia.org/wiki/Causal_set


https://en.wikipedia.org/wiki/Complete_partial_order

https://en.wikipedia.org/wiki/Directed_set

https://en.wikipedia.org/wiki/Graded_poset

https://en.wikipedia.org/wiki/Incidence_algebra


https://en.wikipedia.org/wiki/Locally_finite_poset

https://en.wikipedia.org/wiki/Incidence_algebra

https://en.wikipedia.org/wiki/Ordered_group

44.15. NOTES 183

• poset topology, a kind of topological space that can be defined from any poset

• Scott continuity - continuity of a function between two partial orders.

• semilattice

• semiorder

• series-parallel partial order

• stochastic dominance

• strict weak ordering - strict partial order "<" in which the relation “neither a < b nor b < a" is transitive.

• Zorn’s lemma

44.15 Notes[1] Merrifield, Richard E.; Simmons, Howard E. (1989). Topological Methods in Chemistry. New York: John Wiley & Sons.

p. 28. ISBN 0-471-83817-9. Retrieved 27 July 2012. A partially ordered set is conveniently represented by a Hassediagram...

[2] Simovici, Dan A. & Djeraba, Chabane (2008). “Partially Ordered Sets”. Mathematical Tools for Data Mining: Set Theory,Partial Orders, Combinatorics. Springer. ISBN 9781848002012.

[3] Neggers, J.; Kim, Hee Sik (1998), “4.2 Product Order and Lexicographic Order”, Basic Posets, World Scientific, pp. 62–63,ISBN 9789810235895

[4] Davey & Priestley, Introduction to Lattices and Order (Second Edition), 2002, p. 17-18

[5] P. R. Halmos (1974). Naive Set Theory. Springer. p. 82. ISBN 978-1-4757-1645-0.


[7] Davey, B. A.; Priestley, H. A. (2002). “Maps between ordered sets”. Introduction to Lattices and Order (2nd ed.). NewYork: Cambridge University Press. pp. 23–24. ISBN 0-521-78451-4. MR 1902334.

[8] Jech, Thomas (2008) [1973]. The Axiom of Choice. Dover Publications. ISBN 0-486-46624-8.

[9] Ward, L. E. Jr (1954). “Partially Ordered Topological Spaces”. Proceedings of the American Mathematical Society 5 (1):144–161. doi:10.1090/S0002-9939-1954-0063016-5

44.16 References• Deshpande, Jayant V. (1968). “On Continuity of a Partial Order”. Proceedings of the American Mathematical

Society 19 (2): 383–386. doi:10.1090/S0002-9939-1968-0236071-7.

• Schröder, Bernd S. W. (2003). Ordered Sets: An Introduction. Birkhäuser, Boston.

• Stanley, Richard P.. Enumerative Combinatorics 1. Cambridge Studies in Advanced Mathematics 49. Cam-bridge University Press. ISBN 0-521-66351-2.

44.17 External links• A001035: Number of posets with n labeled elements in the OEIS

• A000112: Number of posets with n unlabeled elements in the OEIS

https://en.wikipedia.org/wiki/Poset_topology

https://en.wikipedia.org/wiki/Scott_continuity


https://en.wikipedia.org/wiki/Semiorder

https://en.wikipedia.org/wiki/Series-parallel_partial_order

https://en.wikipedia.org/wiki/Stochastic_dominance


https://en.wikipedia.org/wiki/Zorn%2527s_lemma

https://en.wikipedia.org/wiki/Howard_Ensign_Simmons,_Jr.

http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471838179.html



http://books.google.com/books?id=6i-F3ZNcub4C&pg=PA127

http://books.google.com/books?id=6i-F3ZNcub4C&pg=PA127







http://www.karlin.mff.cuni.cz/~jezek/120/transitive1.pdf?

http://books.google.com/books?id=vVVTxeuiyvQC&pg=PA23





https://en.wikipedia.org/wiki/Thomas_Jech

https://en.wikipedia.org/wiki/Dover_Publications




https://dx.doi.org/10.1090%252FS0002-9939-1954-0063016-5


https://dx.doi.org/10.1090%252FS0002-9939-1968-0236071-7

https://en.wikipedia.org/wiki/Richard_P._Stanley






Chapter 45

Preorder

Not to be confused with Pre-order.This article is about binary relations. For the graph vertex ordering, see Depth-first search. For other uses, seePreorder (disambiguation).“Quasiorder” redirects here. For irreflexive transitive relations, see strict order.

In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive.All partial orders and equivalence relations are preorders, but preorders are more general.The name 'preorder' comes from the idea that preorders (that are not partial orders) are 'almost' (partial) orders,but not quite; they're neither necessarily anti-symmetric nor symmetric. Because a preorder is a binary relation,the symbol ≤ can be used as the notational device for the relation. However, because they are not necessarily anti-symmetric, some of the ordinary intuition associated to the symbol ≤ may not apply. On the other hand, a preordercan be used, in a straightforward fashion, to define a partial order and an equivalence relation. Doing so, however, isnot always useful or worthwhile, depending on the problem domain being studied.In words, when a ≤ b, one may say that b covers a or that a precedes b, or that b reduces to a. Occasionally, thenotation ← or ≲ is used instead of ≤.To every preorder, there corresponds a directed graph, with elements of the set corresponding to vertices, and theorder relation between pairs of elements corresponding to the directed edges between vertices. The converse is nottrue: most directed graphs are neither reflexive nor transitive. Note that, in general, the corresponding graphs maycontain cycles. A preorder that is antisymmetric no longer has cycles; it is a partial order, and corresponds to adirected acyclic graph. A preorder that is symmetric is an equivalence relation; it can be thought of as having lost thedirection markers on the edges of the graph. In general, a preorder may have many disconnected components.


Consider some set P and a binary relation ≤ on P. Then ≤ is a preorder, or quasiorder, if it is reflexive and transitive,i.e., for all a, b and c in P, we have that:

a ≤ a (reflexivity)if a ≤ b and b ≤ c then a ≤ c (transitivity)

A set that is equipped with a preorder is called a preordered set (or proset).[1]

If a preorder is also antisymmetric, that is, a ≤ b and b ≤ a implies a = b, then it is a partial order.On the other hand, if it is symmetric, that is, if a ≤ b implies b ≤ a, then it is an equivalence relation.Equivalently, the notion of a preordered set P can be formulated in a categorical framework as a thin category, i.e.as a category with at most one morphism from an object to another. Here the objects correspond to the elementsof P, and there is one morphism for objects which are related, zero otherwise. Alternately, a preordered set can beunderstood as an enriched category, enriched over the category 2 = (0→1).

184

https://en.wikipedia.org/wiki/Pre-order

https://en.wikipedia.org/wiki/Depth-first_search

https://en.wikipedia.org/wiki/Preorder_(disambiguation)









https://en.wikipedia.org/wiki/Anti-symmetric_relation














https://en.wikipedia.org/wiki/Thin_category

https://en.wikipedia.org/wiki/Object_(category_theory)

https://en.wikipedia.org/wiki/Enriched_category

45.2. EXAMPLES 185

A preordered class is a class equipped with a preorder. Every set is a class and so every preordered set is a preorderedclass.

45.2 Examples• The reachability relationship in any directed graph (possibly containing cycles) gives rise to a preorder, where

x ≤ y in the preorder if and only if there is a path from x to y in the directed graph. Conversely, every preorderis the reachability relationship of a directed graph (for instance, the graph that has an edge from x to y for everypair (x, y) with x ≤ y). However, many different graphs may have the same reachability preorder as each other.In the same way, reachability of directed acyclic graphs, directed graphs with no cycles, gives rise to partiallyordered sets (preorders satisfying an additional anti-symmetry property).

• Every finite topological space gives rise to a preorder on its points, in which x ≤ y if and only if x belongs to everyneighborhood of y, and every finite preorder can be formed as the specialization preorder of a topological spacein this way. That is, there is a 1-to-1 correspondence between finite topologies and finite preorders. However,the relation between infinite topological spaces and their specialization preorders is not 1-to-1.

• A net is a directed preorder, that is, each pair of elements has an upper bound. The definition of convergencevia nets is important in topology, where preorders cannot be replaced by partially ordered sets without losingimportant features.

• The relation defined by x ≤ y if f(x) ≤ f(y) , where f is a function into some preorder.

• The relation defined by x ≤ y if there exists some injection from x to y. Injection may be replaced by surjection,or any type of structure-preserving function, such as ring homomorphism, or permutation.

• The embedding relation for countable total orderings.

• The graph-minor relation in graph theory.

• A category with at most one morphism from any object x to any other object y is a preorder. Such categoriesare called thin. In this sense, categories “generalize” preorders by allowing more than one relation betweenobjects: each morphism is a distinct (named) preorder relation.

In computer science, one can find examples of the following preorders.

• Many-one and Turing reductions are preorders on complexity classes.

• The subtyping relations are usually preorders.

• Simulation preorders are preorders (hence the name).

• Reduction relations in abstract rewriting systems.

• The encompassment preorder on the set of terms, defined by s≤t if a subterm of t is a substitution instance ofs.

Example of a total preorder:

• Preference, according to common models.

45.3 Uses

Preorders play a pivotal role in several situations:

• Every preorder can be given a topology, the Alexandrov topology; and indeed, every preorder on a set is inone-to-one correspondence with an Alexandrov topology on that set.

• Preorders may be used to define interior algebras.

• Preorders provide the Kripke semantics for certain types of modal logic.

https://en.wikipedia.org/wiki/Preordered_class







https://en.wikipedia.org/wiki/Finite_topological_space

https://en.wikipedia.org/wiki/Specialization_(pre)order


https://en.wikipedia.org/wiki/Net_(mathematics)







https://en.wikipedia.org/wiki/Ring_homomorphism




https://en.wikipedia.org/wiki/Graph-minor




https://en.wikipedia.org/wiki/Thin_category

https://en.wikipedia.org/wiki/Many-one_reduction

https://en.wikipedia.org/wiki/Turing_reduction

https://en.wikipedia.org/wiki/Subtype

https://en.wikipedia.org/wiki/Simulation_preorder

https://en.wikipedia.org/wiki/Reduction_relation


https://en.wikipedia.org/wiki/Encompassment_preorder

https://en.wikipedia.org/wiki/Term_(logic)

https://en.wikipedia.org/wiki/Term_(logic)#Operations_with_terms

https://en.wikipedia.org/wiki/Substitution_instance

https://en.wikipedia.org/wiki/Strict_weak_ordering#Total_preorders


https://en.wikipedia.org/wiki/Alexandrov_topology

https://en.wikipedia.org/wiki/Interior_algebra

https://en.wikipedia.org/wiki/Kripke_semantics


186 CHAPTER 45. PREORDER

45.4 Constructions

Every binary relation R on a set S can be extended to a preorder on S by taking the transitive closure and reflexiveclosure, R+=. The transitive closure indicates path connection in R: x R+ y if and only if there is an R-path from x toy.Given a preorder ≲ on S one may define an equivalence relation ~ on S such that a ~ b if and only if a ≲ b and b ≲a. (The resulting relation is reflexive since a preorder is reflexive, transitive by applying transitivity of the preordertwice, and symmetric by definition.)Using this relation, it is possible to construct a partial order on the quotient set of the equivalence, S / ~, the set ofall equivalence classes of ~. Note that if the preorder is R+=, S / ~ is the set of R-cycle equivalence classes: x ∈ [y]if and only if x = y or x is in an R-cycle with y. In any case, on S / ~ we can define [x] ≤ [y] if and only if x ≲ y.By the construction of ~, this definition is independent of the chosen representatives and the corresponding relationis indeed well-defined. It is readily verified that this yields a partially ordered set.Conversely, from a partial order on a partition of a set S one can construct a preorder on S. There is a 1-to-1 corre-spondence between preorders and pairs (partition, partial order).For a preorder " ≲ ", a relation "<" can be defined as a < b if and only if (a≲ b and not b≲ a), or equivalently, usingthe equivalence relation introduced above, (a ≲ b and not a ~ b). It is a strict partial order; every strict partial ordercan be the result of such a construction. If the preorder is anti-symmetric, hence a partial order "≤", the equivalenceis equality, so the relation "<" can also be defined as a < b if and only if (a ≤ b and a ≠ b).(Alternatively, for a preorder " ≲ ", a relation "<" can be defined as a < b if and only if (a ≲ b and a ≠ b). The resultis the reflexive reduction of the preorder. However, if the preorder is not anti-symmetric the result is not transitive,and if it is, as we have seen, it is the same as before.)Conversely we have a ≲ b if and only if a < b or a ~ b. This is the reason for using the notation " ≲ "; "≤" can beconfusing for a preorder that is not anti-symmetric, it may suggest that a ≤ b implies that a < b or a = b.Note that with this construction multiple preorders " ≲ " can give the same relation "<", so without more information,such as the equivalence relation, " ≲ " cannot be reconstructed from "<". Possible preorders include the following:

• Define a ≤ b as a < b or a = b (i.e., take the reflexive closure of the relation). This gives the partial orderassociated with the strict partial order "<" through reflexive closure; in this case the equivalence is equality, sowe don't need the notations ≲ and ~.

• Define a≲ b as “not b < a" (i.e., take the inverse complement of the relation), which corresponds to defining a~ b as “neither a < b nor b < a"; these relations ≲ and ~ are in general not transitive; however, if they are, ~ isan equivalence; in that case "<" is a strict weak order. The resulting preorder is total, that is, a total preorder.

45.5 Number of preorders

As explained above, there is a 1-to-1 correspondence between preorders and pairs (partition, partial order). Thus thenumber of preorders is the sum of the number of partial orders on every partition. For example:

• for n=3:

• 1 partition of 3, giving 1 preorder• 3 partitions of 2+1, giving 3 × 3 = 9 preorders• 1 partition of 1+1+1, giving 19 preorders

i.e. together 29 preorders.

• for n=4:

• 1 partition of 4, giving 1 preorder• 7 partitions with two classes (4 of 3+1 and 3 of 2+2), giving 7 × 3 = 21 preorders• 6 partitions of 2+1+1, giving 6 × 19 = 114 preorders




https://en.wikipedia.org/wiki/Path_(graph_theory)







https://en.wikipedia.org/wiki/Total_preorder

45.6. INTERVAL 187

• 1 partition of 1+1+1+1, giving 219 preorders

i.e. together 355 preorders.

45.6 Interval

For a ≲ b, the interval [a,b] is the set of points x satisfying a ≲ x and x ≲ b, also written a ≲ x ≲ b. It contains atleast the points a and b. One may choose to extend the definition to all pairs (a,b). The extra intervals are all empty.Using the corresponding strict relation "<", one can also define the interval (a,b) as the set of points x satisfying a <x and x < b, also written a < x < b. An open interval may be empty even if a < b.Also [a,b) and (a,b] can be defined similarly.

45.7 See also• partial order - preorder that is antisymmetric

• equivalence relation - preorder that is symmetric

• total preorder - preorder that is total

• total order - preorder that is antisymmetric and total

• directed set

• category of preordered sets

• prewellordering

• Well-quasi-ordering

45.8 References[1] For “proset”, see e.g. Eklund, Patrik; Gähler, Werner (1990), “Generalized Cauchy spaces”, Mathematische Nachrichten

147: 219–233, doi:10.1002/mana.19901470123, MR 1127325.

• Schröder, Bernd S. W. (2002), Ordered Sets: An Introduction, Boston: Birkhäuser, ISBN 0-8176-4128-9










https://en.wikipedia.org/wiki/Category_of_preordered_sets

https://en.wikipedia.org/wiki/Prewellordering

https://en.wikipedia.org/wiki/Well-quasi-ordering


https://dx.doi.org/10.1002%252Fmana.19901470123





Chapter 46

Prewellordering

In set theory, a prewellordering is a binary relation ≤ that is transitive, total, and wellfounded (more precisely, therelation x ≤ y ∧ y ≰ x is wellfounded). In other words, if ≤ is a prewellordering on a set X , and if we define ∼ by

x ∼ y ⇐⇒ x ≤ y ∧ y ≤ x

then ∼ is an equivalence relation on X , and ≤ induces a wellordering on the quotient X/ ∼ . The order-type of thisinduced wellordering is an ordinal, referred to as the length of the prewellordering.A norm on a set X is a map from X into the ordinals. Every norm induces a prewellordering; if ϕ : X → Ord is anorm, the associated prewellordering is given by

x ≤ y ⇐⇒ ϕ(x) ≤ ϕ(y)

Conversely, every prewellordering is induced by a unique regular norm (a norm ϕ : X → Ord is regular if, for anyx ∈ X and any α < ϕ(x) , there is y ∈ X such that ϕ(y) = α ).

46.1 Prewellordering property

If Γ is a pointclass of subsets of some collection F of Polish spaces, F closed under Cartesian product, and if ≤ is aprewellordering of some subset P of some element X of F , then ≤ is said to be a Γ -prewellordering of P if therelations <∗ and ≤∗ are elements of Γ , where for x, y ∈ X ,

1. x <∗ y ⇐⇒ x ∈ P ∧ [y /∈ P ∨ {x ≤ y ∧ y ̸≤ x}]

2. x ≤∗ y ⇐⇒ x ∈ P ∧ [y /∈ P ∨ x ≤ y]

Γ is said to have the prewellordering property if every set in Γ admits a Γ -prewellordering.The prewellordering property is related to the stronger scale property; in practice, many pointclasses having theprewellordering property also have the scale property, which allows drawing stronger conclusions.

46.1.1 Examples

Π11 andΣ1

2 both have the prewellordering property; this is provable in ZFC alone. Assuming sufficient large cardinals,for every n ∈ ω , Π1

2n+1 and Σ12n+2 have the prewellordering property.

46.1.2 Consequences

188





https://en.wikipedia.org/wiki/Well-founded_relation


https://en.wikipedia.org/wiki/Wellordering


https://en.wikipedia.org/wiki/Order-type


https://en.wikipedia.org/wiki/Pointclass

https://en.wikipedia.org/wiki/Polish_space


https://en.wikipedia.org/wiki/Scale_property

https://en.wikipedia.org/wiki/Zermelo-Fraenkel_set_theory

https://en.wikipedia.org/wiki/Large_cardinal

46.2. SEE ALSO 189

Reduction

If Γ is an adequate pointclass with the prewellordering property, then it also has the reduction property: For anyspace X ∈ F and any sets A,B ⊆ X , A and B both in Γ , the union A ∪B may be partitioned into sets A∗, B∗ ,both in Γ , such that A∗ ⊆ A and B∗ ⊆ B .

Separation

If Γ is an adequate pointclass whose dual pointclass has the prewellordering property, then Γ has the separationproperty: For any space X ∈ F and any sets A,B ⊆ X , A and B disjoint sets both in Γ , there is a set C ⊆ Xsuch that both C and its complement X \ C are in Γ , with A ⊆ C and B ∩ C = ∅ .For example, Π1

1 has the prewellordering property, so Σ11 has the separation property. This means that if A and B

are disjoint analytic subsets of some Polish space X , then there is a Borel subset C of X such that C includes A andis disjoint from B .

46.2 See also• Descriptive set theory

• Scale property

• Graded poset – a graded poset is analogous to a prewellordering with a norm, replacing a map to the ordinalswith a map to the integers

46.3 References• Moschovakis, Yiannis N. (1980). Descriptive Set Theory. North Holland. ISBN 0-444-70199-0.

https://en.wikipedia.org/wiki/Adequate_pointclass

https://en.wikipedia.org/wiki/Adequate_pointclass

https://en.wikipedia.org/wiki/Dual_pointclass


https://en.wikipedia.org/wiki/Analytic_set

https://en.wikipedia.org/wiki/Borel_set

https://en.wikipedia.org/wiki/Descriptive_set_theory

https://en.wikipedia.org/wiki/Scale_property

https://en.wikipedia.org/wiki/Graded_poset



Chapter 47

Sequential composition

In computer science, the process calculi (or process algebras) are a diverse family of related approaches for for-mally modelling concurrent systems. Process calculi provide a tool for the high-level description of interactions,communications, and synchronizations between a collection of independent agents or processes. They also providealgebraic laws that allow process descriptions to be manipulated and analyzed, and permit formal reasoning aboutequivalences between processes (e.g., using bisimulation). Leading examples of process calculi include CSP, CCS,ACP, and LOTOS.[1] More recent additions to the family include the π-calculus, the ambient calculus, PEPA, thefusion calculus and the join-calculus.

47.1 Essential features

While the variety of existing process calculi is very large (including variants that incorporate stochastic behaviour,timing information, and specializations for studying molecular interactions), there are several features that all processcalculi have in common:[2]

• Representing interactions between independent processes as communication (message-passing), rather than asmodification of shared variables.

• Describing processes and systems using a small collection of primitives, and operators for combining thoseprimitives.

• Defining algebraic laws for the process operators, which allow process expressions to be manipulated usingequational reasoning.

47.2 Mathematics of processes

To define a process calculus, one starts with a set of names (or channels) whose purpose is to provide means ofcommunication. In many implementations, channels have rich internal structure to improve efficiency, but this isabstracted away in most theoretic models. In addition to names, one needs a means to form new processes from old.The basic operators, always present in some form or other, allow:[3]

• parallel composition of processes

• specification of which channels to use for sending and receiving data

• sequentialization of interactions

• hiding of interaction points

• recursion or process replication

190


https://en.wikipedia.org/wiki/Concurrent_system

https://en.wikipedia.org/wiki/Algebra

https://en.wikipedia.org/wiki/Bisimulation

https://en.wikipedia.org/wiki/Communicating_Sequential_Processes

https://en.wikipedia.org/wiki/Calculus_of_Communicating_Systems

https://en.wikipedia.org/wiki/Algebra_of_Communicating_Processes

https://en.wikipedia.org/wiki/Language_Of_Temporal_Ordering_Specification

https://en.wikipedia.org/wiki/Pi-calculus

https://en.wikipedia.org/wiki/Ambient_calculus

https://en.wikipedia.org/wiki/PEPA

https://en.wikipedia.org/wiki/Fusion_calculus

https://en.wikipedia.org/wiki/Join-calculus

https://en.wikipedia.org/wiki/Stochastic

https://en.wikipedia.org/wiki/Message-passing

https://en.wikipedia.org/wiki/Equational_reasoning

https://en.wikipedia.org/wiki/Channel_(programming)

47.2. MATHEMATICS OF PROCESSES 191

47.2.1 Parallel composition

Parallel composition of two processes P and Q , usually written P |Q , is the key primitive distinguishing the processcalculi from sequential models of computation. Parallel composition allows computation in P and Q to proceedsimultaneously and independently. But it also allows interaction, that is synchronisation and flow of information fromP to Q (or vice versa) on a channel shared by both. Crucially, an agent or process can be connected to more than onechannel at a time.Channels may be synchronous or asynchronous. In the case of a synchronous channel, the agent sending a messagewaits until another agent has received the message. Asynchronous channels do not require any such synchronization.In some process calculi (notably the π-calculus) channels themselves can be sent in messages through (other) channels,allowing the topology of process interconnections to change. Some process calculi also allow channels to be createdduring the execution of a computation.

47.2.2 Communication

Interaction can be (but isn't always) a directed flow of information. That is, input and output can be distinguished asdual interaction primitives. Process calculi that make such distinctions typically define an input operator (e.g. x(v)) and an output operator (e.g. x⟨y⟩ ), both of which name an interaction point (here x ) that is used to synchronisewith a dual interaction primitive.Information should be exchanged, it will flow from the outputting to the inputting process. The output primitive willspecify the data to be sent. In x⟨y⟩ , this data is y . Similarly, if an input expects to receive data, one or more boundvariables will act as place-holders to be substituted by data, when it arrives. In x(v) , v plays that role. The choice ofthe kind of data that can be exchanged in an interaction is one of the key features that distinguishes different processcalculi.

47.2.3 Sequential composition

Sometimes interactions must be temporally ordered. For example, it might be desirable to specify algorithms such as:first receive some data on x and then send that data on y . Sequential composition can be used for such purposes. It iswell known from other models of computation. In process calculi, the sequentialisation operator is usually integratedwith input or output, or both. For example, the process x(v) · P will wait for an input on x . Only when this inputhas occurred will the process P be activated, with the received data through x substituted for identifier v .

47.2.4 Reduction semantics

The key operational reduction rule, containing the computational essence of process calculi, can be given solely interms of parallel composition, sequentialization, input, and output. The details of this reduction vary among thecalculi, but the essence remains roughly the same. The reduction rule is:

x⟨y⟩ · P | x(v) ·Q −→ P | Q[y/v]

The interpretation of this reduction rule is:

1. The process x⟨y⟩ ·P sends a message, here y , along the channel x . Dually, the process x(v) ·Q receives thatmessage on channel x .

2. Once the message has been sent, x⟨y⟩ · P becomes the process P , while x(v) ·Q becomes the process Q[y/v], which is Q with the place-holder v substituted by y , the data received on x .

The class of processes that P is allowed to range over as the continuation of the output operation substantially influ-ences the properties of the calculus.

https://en.wikipedia.org/wiki/%CE%A0-calculus

https://en.wikipedia.org/wiki/Bound_variables


192 CHAPTER 47. SEQUENTIAL COMPOSITION

47.2.5 Hiding

Processes do not limit the number of connections that can be made at a given interaction point. But interaction pointsallow interference (i.e. interaction). For the synthesis of compact, minimal and compositional systems, the ability torestrict interference is crucial. Hiding operations allow control of the connections made between interaction pointswhen composing agents in parallel. Hiding can be denoted in a variety of ways. For example, in the π -calculus thehiding of a name x in P can be expressed as (ν x)P , while in CSP it might be written as P \ {x} .

47.2.6 Recursion and replication

The operations presented so far describe only finite interaction and are consequently insufficient for full computability,which includes non-terminating behaviour. Recursion and replication are operations that allow finite descriptionsof infinite behaviour. Recursion is well known from the sequential world. Replication !P can be understood asabbreviating the parallel composition of a countably infinite number of P processes:

!P = P |!P

47.2.7 Null process

Process calculi generally also include a null process (variously denoted as nil , 0 , STOP , δ , or some other appropriatesymbol) which has no interaction points. It is utterly inactive and its sole purpose is to act as the inductive anchor ontop of which more interesting processes can be generated.

47.3 Discrete and continuous process algebra

Process algebra has been studied for discrete time and continuous time (real time or dense time).[4]

47.4 History

In the first half of the 20th century, various formalisms were proposed to capture the informal concept of a com-putable function, with μ-recursive functions, Turing Machines and the lambda calculus possibly being the best-knownexamples today. The surprising fact that they are essentially equivalent, in the sense that they are all encodable intoeach other, supports the Church-Turing thesis. Another shared feature is more rarely commented on: they all aremost readily understood as models of sequential computation. The subsequent consolidation of computer science re-quired a more subtle formulation of the notion of computation, in particular explicit representations of concurrencyand communication. Models of concurrency such as the process calculi, Petri nets in 1962, and the Actor model in1973 emerged from this line of enquiry.Research on process calculi began in earnest with Robin Milner's seminal work on the Calculus of CommunicatingSystems (CCS) during the period from 1973 to 1980. C.A.R. Hoare's Communicating Sequential Processes (CSP)first appeared in 1978, and was subsequently developed into a full-fledged process calculus during the early 1980s.There was much cross-fertilization of ideas between CCS and CSP as they developed. In 1982 Jan Bergstra andJan Willem Klop began work on what came to be known as the Algebra of Communicating Processes (ACP), andintroduced the term process algebra to describe their work.[1] CCS, CSP, and ACP constitute the three major branchesof the process calculi family: the majority of the other process calculi can trace their roots to one of these three calculi.

47.5 Current research

Various process calculi have been studied and not all of them fit the paradigm sketched here. The most prominentexample may be the ambient calculus. This is to be expected as process calculi are an active field of study. Currentlyresearch on process calculi focuses on the following problems.

https://en.wikipedia.org/wiki/Communicating_sequential_processes

https://en.wikipedia.org/wiki/%CE%9C-recursive_function

https://en.wikipedia.org/wiki/Turing_Machine

https://en.wikipedia.org/wiki/Lambda_calculus

https://en.wikipedia.org/wiki/Church-Turing_thesis

https://en.wikipedia.org/wiki/Petri_net

https://en.wikipedia.org/wiki/Actor_model

https://en.wikipedia.org/wiki/Robin_Milner



https://en.wikipedia.org/wiki/C.A.R._Hoare


https://en.wikipedia.org/wiki/Jan_Bergstra

https://en.wikipedia.org/wiki/Jan_Willem_Klop



47.6. SOFTWARE IMPLEMENTATIONS 193

• Developing new process calculi for better modeling of computational phenomena.

• Finding well-behaved subcalculi of a given process calculus. This is valuable because (1) most calculi are fairlywild in the sense that they are rather general and not much can be said about arbitrary processes; and (2)computational applications rarely exhaust the whole of a calculus. Rather they use only processes that are veryconstrained in form. Constraining the shape of processes is mostly studied by way of type systems.

• Logics for processes that allow one to reason about (essentially) arbitrary properties of processes, following theideas of Hoare logic.

• Behavioural theory: what does it mean for two processes to be the same? How can we decide whether twoprocesses are different or not? Can we find representatives for equivalence classes of processes? Generally,processes are considered to be the same if no context, that is other processes running in parallel, can detect adifference. Unfortunately, making this intuition precise is subtle and mostly yields unwieldy characterisations ofequality (which in most cases must also be undecidable, as a consequence of the halting problem). Bisimulationsare a technical tool that aids reasoning about process equivalences.

• Expressivity of calculi. Programming experience shows that certain problems are easier to solve in somelanguages than in others. This phenomenon calls for a more precise characterisation of the expressivity ofcalculi modeling computation than that afforded by the Church-Turing thesis. One way of doing this is toconsider encodings between two formalisms and see what properties encodings can potentially preserve. Themore properties can be preserved, the more expressive the target of the encoding is said to be. For processcalculi, the celebrated results are that the synchronous π -calculus is more expressive than its asynchronousvariant, has the same expressive power as the higher-order π -calculus, but is less than the ambient calculus.

• Using process calculus to model biological systems (stochasticπ -calculus, BioAmbients, Beta Binders, BioPEPA,Brane calculus). It is thought by some that the compositionality offered by process-theoretic tools can help bi-ologists to organise their knowledge more formally.

47.6 Software implementations

The ideas behind process algebra have given rise to several tools including:

• CADP

• Concurrency Workbench

• mCRL2 toolset

47.7 Relationship to other models of concurrency

The history monoid is the free object that is generically able to represent the histories of individual communicatingprocesses. A process calculus is then a formal language imposed on a history monoid in a consistent fashion.[5] Thatis, a history monoid can only record a sequence of events, with synchronization, but does not specify the allowedstate transitions. Thus, a process calculus is to a history monoid what a formal language is to a free monoid (a formallanguage is a subset of the set of all possible finite-length strings of an alphabet generated by the Kleene star).The use of channels for communication is one of the features distinguishing the process calculi from other models ofconcurrency, such as Petri nets and the Actor model (see Actor model and process calculi). One of the fundamentalmotivations for including channels in the process calculi was to enable certain algebraic techniques, thereby makingit easier to reason about processes algebraically.

47.8 See also• Stochastic probe


https://en.wikipedia.org/wiki/Hoare_logic

https://en.wikipedia.org/wiki/Halting_problem




https://en.wikipedia.org/wiki/Compositionality

https://en.wikipedia.org/wiki/CADP

http://homepages.inf.ed.ac.uk/perdita/cwb

http://www.mcrl2.org/

https://en.wikipedia.org/wiki/History_monoid


https://en.wikipedia.org/wiki/Formal_language




https://en.wikipedia.org/wiki/Concurrent_computing



https://en.wikipedia.org/wiki/Actor_model_and_process_calculi

https://en.wikipedia.org/wiki/Stochastic_probe


47.9 References[1] Baeten, J.C.M. (2004). “A brief history of process algebra” (PDF). Rapport CSR 04-02 (Vakgroep Informatica, Technische

Universiteit Eindhoven).

[2] Pierce, Benjamin. “Foundational Calculi for Programming Languages”. The Computer Science and Engineering Handbook.CRC Press. pp. 2190–2207. ISBN 0-8493-2909-4.

[3] Baeten, J.C.M.; Bravetti, M. (August 2005). “A Generic Process Algebra”. Algebraic Process Calculi: The First TwentyFive Years and Beyond (BRICS Notes Series NS-05-3). Bertinoro, Forl`ı, Italy: BRICS, Department of Computer Science,University of Aarhus. Retrieved 2007-12-29.

[4] Baeten, J. C. M.; Middelburg, C. A. “Process algebra with timing: Real time and discrete time”. CiteSeerX: 10 .1 .1 .42 .729.

[5] Mazurkiewicz, Antoni (1995). “Introduction to Trace Theory”. In Diekert, V.; Rozenberg, G. The Book of Traces(POSTSCRIPT). Singapore: World Scientific. pp. 3–41. ISBN 981-02-2058-8.

47.10 Further reading• Matthew Hennessy: Algebraic Theory of Processes, The MIT Press, ISBN 0-262-08171-7.

• C. A. R. Hoare: Communicating Sequential Processes, Prentice Hall, ISBN 0-13-153289-8.

• This book has been updated by Jim Davies at the Oxford University Computing Laboratory and the newedition is available for download as a PDF file at the Using CSP website.

• Robin Milner: A Calculus of Communicating Systems, Springer Verlag, ISBN 0-387-10235-3.

• Robin Milner: Communicating and Mobile Systems: the Pi-Calculus, Springer Verlag, ISBN 0-521-65869-1.

• Andrew Mironov: Theory of processes

http://alexandria.tue.nl/extra1/wskrap/publichtml/200402.pdf

https://en.wikipedia.org/wiki/Benjamin_C._Pierce



http://www.brics.dk/NS/05/3/



http://www.ipipan.waw.pl/~amaz/papers.htm/trbook.ps



https://en.wikipedia.org/wiki/Matthew_Hennessy

https://en.wikipedia.org/wiki/The_MIT_Press


https://en.wikipedia.org/wiki/C._A._R._Hoare

https://en.wikipedia.org/wiki/Prentice_Hall


https://en.wikipedia.org/wiki/Oxford_University_Computing_Laboratory

https://en.wikipedia.org/wiki/Portable_Document_Format

http://www.usingcsp.com/





https://en.wikipedia.org/wiki/Andrew_Mironov

http://arxiv.org/abs/1009.2259

Chapter 48

Propositional function

A propositional function in logic, is a sentence expressed in a way that would assume the value of true or false, exceptthat within the sentence is a variable (x) that is not defined or specified, which leaves the statement undetermined. Ofcourse, x could also consist of several variables.As a mathematical function, A(x) or A(x1, x2, · · ·, xn), the propositional function is abstracted from predicates orpropositional forms. As an example, let’s imagine the predicate, “x is hot”. The substitution of any entity for x willproduce a specific proposition that can be described as either true or false, even though "x is hot” on its own has novalue as either a true or false statement. However, when you assign x a value, such as lava, the function then has thevalue true; while if you assign x a value like ice, the function then has the value false.Propositional functions are useful in set theory for the formation of sets. For example, in 1903 Bertrand Russell wrotein The Principles of Mathematics (page 106):

"...it has become necessary to take propositional function as a primitive notion.

Later Russell examined the problem of whether propositional functions were predicative or not, and he proposed twotheories to try to get at this question: the zig-zag theory and the ramified theory of types.[1]

A Propositional Function, or a predicate, in a variable x is a sentence p(x) involving x that becomes a propositionwhen we give x a definite value from the set of values it can take.

48.1 References[1] Tiles, Mary (2004). The philosophy of set theory an historical introduction to Cantor’s paradise (Dover ed.). Mineola, N.Y.:

Dover Publications. p. 159. ISBN 978-0-486-43520-6. Retrieved 1 February 2013.

195


https://en.wikipedia.org/wiki/Truth

https://en.wikipedia.org/wiki/Contradiction

https://en.wikipedia.org/wiki/Variable_(mathematics)



https://en.wikipedia.org/wiki/Lava

https://en.wikipedia.org/wiki/Ice




https://en.wikipedia.org/wiki/The_Principles_of_Mathematics

https://en.wikipedia.org/wiki/Primitive_notion

https://en.wikipedia.org/wiki/Mary_Tiles

http://store.doverpublications.com/0486435202.html



Chapter 49

Quasi-commutative property

In mathematics, the quasi-commutative property is an extension or generalization of the general commutativeproperty. This property is used in certain specific applications with various definitions.

49.1 Applied to matrices

Two matrices p and q are said to have the commutative property whenever

pq = qp

The quasi-commutative property in matrices is defined[1] as follows. Given two non-commutable matrices x and y

xy − yx = z

satisfy the quasi-commutative property whenever z satisfies the following properties:

xz = zx

yz = zy

An example is found in the matrix mechanics introduced by Heisenberg as a version of quantum mechanics. In thismechanics, p and q are infinite matrices corresponding respectively to the momentum and position variables of aparticle.[1] These matrices are written out at Matrix mechanics#Harmonic oscillator, and z = iħ times the infinite unitmatrix, where ħ is the reduced Planck constant.

49.2 Applied to functions

A function f, defined as follows:

f : X × Y → X

is said to be quasi-commutative[2] if for all x ∈ X and for all y1, y2 ∈ Y ,

f(f(x, y1), y2) = f(f(x, y2), y1)

196


https://en.wikipedia.org/wiki/Commutative_property




https://en.wikipedia.org/wiki/Matrix_mechanics

https://en.wikipedia.org/wiki/Werner_Heisenberg

https://en.wikipedia.org/wiki/Quantum_mechanics

https://en.wikipedia.org/wiki/Matrix_mechanics#Harmonic_oscillator

https://en.wikipedia.org/wiki/Unit_matrix

https://en.wikipedia.org/wiki/Unit_matrix

https://en.wikipedia.org/wiki/Reduced_Planck_constant

49.3. SEE ALSO 197

49.3 See also• Commutative property

• Accumulator (cryptography)

49.4 References[1] Neal H. McCoy. On quasi-commutative matrices. Transactions of the American Mathematical Society, 36(2), 327–340.

[2] Benaloh, J., & De Mare, M. (1994, January). One-way accumulators: A decentralized alternative to digital signatures. InAdvances in Cryptology—EUROCRYPT’93 (pp. 274–285). Springer Berlin Heidelberg.


https://en.wikipedia.org/wiki/Accumulator_(cryptography)

http://www.ams.org/journals/tran/1934-036-02/S0002-9947-1934-1501746-8/S0002-9947-1934-1501746-8.pdf

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.13.8287&rep=rep1&type=pdf

Chapter 50

Quasitransitive relation

Quasitransitivity is a weakened version of transitivity that is used in social choice theory or microeconomics. In-formally, a relation is quasitransitive if it is symmetric for some values and transitive elsewhere. The concept wasintroduced by Sen (1969) to study the consequences of Arrow’s theorem.


A binary relation T over a set X is quasitransitive if for all a, b, and c in X the following holds:

(aT b) ∧ ¬(bT a) ∧ (bT c) ∧ ¬(cT b) ⇒ (aT c) ∧ ¬(cT a).

If the relation is also antisymmetric, T is transitive.Alternately, for a relation T, define the asymmetric or “strict” part P:

(aP b) ⇔ (aT b) ∧ ¬(bT a).

Then T is quasitransitive iff P is transitive.

50.2 Examples

Preferences are assumed to be quasitransitive (rather than transitive) in some economic contexts. The classic exampleis a person indifferent between 10 and 11 grams of sugar and indifferent between 11 and 12 grams of sugar, but whoprefers 12 grams of sugar to 10. Similarly, the Sorites paradox can be resolved by weakening assumed transitivity ofcertain relations to quasitransitivity.

50.3 Properties

• Every transitive relation is quasitransitive; every quasitransitive relation is an acyclic relation. In each case theconverse does not hold in general.

50.4 See also

• Intransitivity


198


https://en.wikipedia.org/wiki/Social_choice_theory

https://en.wikipedia.org/wiki/Microeconomics

https://en.wikipedia.org/wiki/Quasitransitive_relation#CITEREFSen1969

https://en.wikipedia.org/wiki/Arrow%2527s_theorem





https://en.wikipedia.org/wiki/Sorites_paradox#Dropping_transitivity_of_involved_relations

https://en.wikipedia.org/wiki/Acyclic_relation




50.5 References• Bossert, Walter; Suzumura, Kōtarō (2010). Consistency, choice and rationality. Harvard University Press.

ISBN 0674052994.

• Sen, A. (1969). “Quasi-transitivity, rational choice and collective decisions”. Rev. Econ. Stud. 36: 381–393.doi:10.2307/2296434. Zbl 0181.47302.



https://en.wikipedia.org/wiki/Amartya_Sen


https://dx.doi.org/10.2307%252F2296434



Chapter 51

Quotient by an equivalence relation

This article is about a generalization to category theory, used in scheme theory. For the common meaning, seeEquivalence class.

In mathematics, given a category C, a quotient of an object X by an equivalence relation f : R → X × X is acoequalizer for the pair of maps

Rf→X ×X

pri→X, i = 1, 2,

where R is an object in C and "f is an equivalence relation” means that, for any object T in C, the image (which isa set) of f : R(T ) = Mor(T,R) → X(T ) × X(T ) is an equivalence relation; that is, (x, y) is in it if and only if(y, x) is in it, etc.The basic case in practice is when C is the category of all schemes over some scheme S. But the notion is flexible andone can also take C to be the category of sheaves.

51.1 Examples

• Let X be a set and consider some equivalence relation on it. Let Q be the set of all equivalence classes in X.Then the map q : X → Q that sends an element x to an equivalence class to which x belong is a quotient.

• In the above example, Q is a subset of the power set H of X. In algebraic geometry, one might replace Hby a Hilbert scheme or disjoint union of Hilbert schemes. In fact, Grothendieck constructed a relative Picardscheme of a flat projective scheme X[1] as a quotient Q (of the scheme Z parametrizing relative effective divisorson X) that is a closed scheme of a Hilbert scheme H. The quotient map q : Z → Q can then be thought of asa relative version of the Abel map.

51.2 See also

• categorical quotient, a special case

51.3 Notes

[1] One also needs to assume the geometric fibers are integral schemes; Mumford’s example shows the “integral” cannot beomitted.

200


https://en.wikipedia.org/wiki/Scheme_theory


https://en.wikipedia.org/wiki/Coequalizer


https://en.wikipedia.org/wiki/Sheaf_(mathematics)


https://en.wikipedia.org/wiki/Hilbert_scheme

https://en.wikipedia.org/wiki/Picard_scheme

https://en.wikipedia.org/wiki/Picard_scheme

https://en.wikipedia.org/wiki/Relative_effective_divisor

https://en.wikipedia.org/wiki/Abel_map

https://en.wikipedia.org/wiki/Categorical_quotient


51.4 References• Nitsure, N. Construction of Hilbert and Quot schemes. Fundamental algebraic geometry: Grothendieck’s FGA

explained, Mathematical Surveys and Monographs 123, American Mathematical Society 2005, 105–137.

Chapter 52

Rational consequence relation

In logic, a rational consequence relation is a non-monotonic consequence relation satisfying certain properties listedbelow.

52.1 Properties

A rational consequence relation satisfies:

REF Reflexivity θ ⊢ θ

and the so-called Gabbay-Makinson rules:

LLE Left Logical Equivalence θ⊢ψ θ≡ϕϕ⊢ψ

RWE Right-hand weakening θ⊢ϕ ϕ|=ψθ⊢ψ

CMO Cautious monotonicity θ⊢ϕ θ⊢ψθ∧ψ⊢ϕ

DIS Logical or (ie disjunction) on left hand side θ⊢ψ ϕ⊢ψθ∨ϕ⊢ψ

AND Logical and on right hand side θ⊢ϕ θ⊢ψθ⊢ϕ∧ψ

RMO Rational monotonicity ϕ̸⊢¬θ ϕ⊢ψϕ∧θ⊢ψ

52.2 Uses

The rational consequence relation is non-monotonic, and the relation θ ⊢ ϕ is intended to carry the meaning thetausually implies phi or phi usually follows from theta. In this sense it is more useful for modeling some everydaysituations than a monotone consequence relation because the latter relation models facts in a more strict booleanfashion - something either follows under all circumstances or it does not.

52.2.1 Example

The statement “If a cake contains sugar then it tastes good” implies under a monotone consequence relation the state-ment “If a cake contains sugar and soap then it tastes good.” Clearly this doesn't match our own understanding ofcakes. By asserting “If a cake contains sugar then it usually tastes good” a rational consequence relation allows fora more realistic model of the real world, and certainly it does not automatically follow that “If a cake contains sugarand soap then it usually tastes good.”

Note that if we also have the information “If a cake contains sugar then it usually contains butter” then we may legallyconclude (under CMO) that “If a cake contains sugar and butter then it usually tastes good.”. Equally in the absence

202


https://en.wikipedia.org/wiki/Non-monotonic_logic

https://en.wikipedia.org/wiki/Consequence_relation


https://en.wikipedia.org/wiki/Monotonic_logic

https://en.wikipedia.org/wiki/Logical_or

https://en.wikipedia.org/wiki/Logical_and

https://en.wikipedia.org/wiki/Monotonic_logic

https://en.wikipedia.org/wiki/Monotone_consequence_relation

52.3. CONSEQUENCES 203

of a statement such as “If a cake contains sugar then usually it contains no soap" then we may legally conclude fromRMO that “If the cake contains sugar and soap then it usually tastes good.”

If this latter conclusion seems ridiculous to you then it is likely that you are subconsciously asserting your own pre-conceived knowledge about cakes when evaluating the validity of the statement. That is, from your experience youknow that cakes which contain soap are likely to taste bad so you add to the system your own knowledge such as“Cakes which contain sugar do not usually contain soap.”, even though this knowledge is absent from it. If the conclu-sion seems silly to you then you might consider replacing the word soap with the word eggs to see if it changes yourfeelings.

52.2.2 Example

Consider the sentences:

• Young people are usually happy

• Drug abusers are usually not happy

• Drug abusers are usually young

We may consider it reasonable to conclude:

• Young drug abusers are usually not happy

This would not be a valid conclusion under a monotonic deduction system (omitting of course the word 'usually'),since the third sentence would contradict the first two. In contrast the conclusion follows immediately using theGabbay-Makinson rules: applying the rule CMO to the last two sentences yields the result.

52.3 Consequences

The following consequences follow from the above rules:

MP Modus ponens θ⊢ϕ θ⊢(ϕ→ψ)θ⊢ψ

MP is proved via the rules AND and RWE.

CON Conditionalisation θ∧ϕ⊢ψθ⊢(ϕ→ψ)

CC Cautious Cut θ⊢ϕ θ∧ϕ⊢ψθ⊢ψ

The notion of Cautious Cut simply encapsulates the operation of conditionalisation, followed by MP.It may seem redundant in this sense, but it is often used in proofs so it is useful to have a name forit to act as a shortcut.

SCL Supraclassity θ|=ϕθ⊢ϕ

SCL is proved trivially via REF and RWE.

52.4 Rational consequence relations via atom preferences

Let L = {p1, . . . , pn} be a finite language. An atom is a formula of the form∧ni=1 p

ϵi (where p1 = p and p−1 = ¬p

). Notice that there is a unique valuation which makes any given atom true (and conversely each valuation satisfiesprecisely one atom). Thus an atom can be used to represent a preference about what we believe ought to be true.Let AtL be the set of all atoms in L. For θ ∈ SL, define Sθ = {α ∈ AtL|α |=SC θ} .Let s⃗ = s1, . . . , sm be a sequence of subsets of AtL . For θ , ϕ in SL, let the relation ⊢s⃗ be such that θ ⊢s⃗ ϕ if oneof the following holds:

https://en.wikipedia.org/wiki/Modus_ponens

https://en.wikipedia.org/wiki/Sentential_logic

204 CHAPTER 52. RATIONAL CONSEQUENCE RELATION

1. Sθ ∩ si = ∅ for each 1 ≤ i ≤ m

2. Sθ ∩ si ̸= ∅ for some 1 ≤ i ≤ m and for the least such i, Sθ ∩ si ⊆ Sϕ .

Then the relation ⊢s⃗ is a rational consequence relation. This may easily be verified by checking directly that it satisfiesthe GM-conditions.The idea behind the sequence of atom sets is that the earlier sets account for the most likely situations such as “youngpeople are usually law abiding” whereas the later sets account for the less likely situations such as “young joyridersare usually not law abiding”.

52.4.1 Notes

1. By the definition of the relation ⊢s⃗ , the relation is unchanged if we replace s2 with s2 \s1 , s3 with s3 \s2 \s1... and sm with sm \

∪m−1i=1 si . In this way we make each si disjoint. Conversely it makes no difference to the

rcr ⊢s⃗ if we add to subsequent si atoms from any of the preceding si .

52.5 The representation theorem

It can be proven that any rational consequence relation on a finite language is representable via a sequence of atompreferences above. That is, for any such rational consequence relation ⊢ there is a sequence s⃗ = s1, . . . , sm of subsetsof AtL such that the associated rcr ⊢s⃗ is the same relation: ⊢s⃗=⊢

52.5.1 Notes

1. By the above property of ⊢s⃗ , the representation of an rcr ⊢ need not be unique - if the si are not disjoint thenthey can be made so without changing the rcr and conversely if they are disjoint then each subsequent set cancontain any of the atoms of the previous sets without changing the rcr.

52.6 References• A mathematical paper in which the GM rules are defined

http://www.maths.manchester.ac.uk/~jeff/theses/lee-hill.ps

Chapter 53

Reduct

This article is about a relation on algebraic structures. For reducts in abstract rewriting, see Confluence (abstractrewriting).

In universal algebra and in model theory, a reduct of an algebraic structure is obtained by omitting some of theoperations and relations of that structure. The converse of “reduct” is “expansion.”

53.1 Definition

Let A be an algebraic structure (in the sense of universal algebra) or equivalently a structure in the sense of modeltheory, organized as a set X together with an indexed family of operations and relations φᵢ on that set, with index setI. Then the reduct of A defined by a subset J of I is the structure consisting of the set X and J-indexed family ofoperations and relations whose j-th operation or relation for j∈J is the j-th operation or relation of A. That is, thisreduct is the structure A with the omission of those operations and relations φi for which i is not in J.A structure A is an expansion of B just when B is a reduct of A. That is, reduct and expansion are mutual converses.

53.2 Examples

The monoid (Z, +, 0) of integers under addition is a reduct of the group (Z, +, −, 0) of integers under addition andnegation, obtained by omitting negation. By contrast, the monoid (N,+,0) of natural numbers under addition is notthe reduct of any group.Conversely the group (Z, +, −, 0) is the expansion of the monoid (Z, +, 0), expanding it with the operation of negation.

53.3 References• Burris, Stanley N.; H. P. Sankappanavar (1981). ACourse in Universal Algebra. Springer. ISBN 3-540-90578-

2.

• Hodges, Wilfrid (1993). Model theory. Cambridge University Press. ISBN 0-521-30442-3.

205

https://en.wikipedia.org/wiki/Confluence_(abstract_rewriting)

https://en.wikipedia.org/wiki/Confluence_(abstract_rewriting)





https://en.wikipedia.org/wiki/Structure_(mathematical_logic)



https://en.wikipedia.org/wiki/Family


https://en.wikipedia.org/wiki/Index_set






http://www.thoralf.uwaterloo.ca/htdocs/ualg.html








Chapter 54

Reflexive closure

In mathematics, the reflexive closure of a binary relation R on a set X is the smallest reflexive relation on X thatcontains R.For example, if X is a set of distinct numbers and x R y means "x is less than y", then the reflexive closure of R is therelation "x is less than or equal to y".

54.1 Definition

The reflexive closure S of a relation R on a set X is given by

S = R ∪ {(x, x) : x ∈ X}

In words, the reflexive closure of R is the union of R with the identity relation on X.

54.2 See also• Transitive closure

• Symmetric closure

54.3 References• Franz Baader and Tobias Nipkow, Term Rewriting and All That, Cambridge University Press, 1998, p. 8

206







https://en.wikipedia.org/wiki/Symmetric_closure

https://en.wikipedia.org/wiki/Franz_Baader

https://en.wikipedia.org/wiki/Tobias_Nipkow

Chapter 55

Reflexive relation

In mathematics, a reflexive relation is a binary relation on a set for which every element is related to itself. In otherwords, a relation ~ on a set S is reflexive when x ~ x holds true for every x in S, formally: when ∀x∈S: x~x holds.[1][2]

An example of a reflexive relation is the relation "is equal to" on the set of real numbers, since every real number isequal to itself. A reflexive relation is said to have the reflexive property or is said to possess reflexivity.

55.1 Related terms

A relation that is irreflexive, or anti-reflexive, is a binary relation on a set where no element is related to itself. Anexample is the “greater than” relation (x>y) on the real numbers. Note that not every relation which is not reflexiveis irreflexive; it is possible to define relations where some elements are related to themselves but others are not (i.e.,neither all nor none are). For example, the binary relation “the product of x and y is even” is reflexive on the set ofeven numbers, irreflexive on the set of odd numbers, and neither reflexive nor irreflexive on the set of natural numbers.A relation ~ on a set S is called quasi-reflexive if every element that is related to some element is also related to itself,formally: if ∀x,y∈S: x~y ⇒ x~x ∧ y~y. An example is the relation “has the same limit as” on the set of sequences ofreal numbers: not every sequence has a limit, and thus the relation is not reflexive, but if a sequence has the samelimit as some sequence, then it has the same limit as itself.The reflexive closure ≃ of a binary relation ~ on a set S is the smallest reflexive relation on S that is a superset of ~.Equivalently, it is the union of ~ and the identity relation on S, formally: (≃) = (~) ∪ (=). For example, the reflexiveclosure of x<y is x≤y.The reflexive reduction, or irreflexive kernel, of a binary relation ~ on a set S is the smallest relation ≆ such that ≆shares the same reflexive closure as ~. It can be seen in a way as the opposite of the reflexive closure. It is equivalentto the complement of the identity relation on S with regard to ~, formally: (≆) = (~) \ (=). That is, it is equivalent to~ except for where x~x is true. For example, the reflexive reduction of x≤y is x<y.

55.2 Examples

Examples of reflexive relations include:

• “is equal to” (equality)

• “is a subset of” (set inclusion)

• “divides” (divisibility)

• “is greater than or equal to”

• “is less than or equal to”

Examples of irreflexive relations include:

207














208 CHAPTER 55. REFLEXIVE RELATION

• “is not equal to”

• “is coprime to” (for the integers>1, since 1 is coprime to itself)

• “is a proper subset of”

• “is greater than”

• “is less than”

55.3 Number of reflexive relations

The number of reflexive relations on an n-element set is 2n2−n.[3]

https://en.wikipedia.org/wiki/Coprime

55.4. PHILOSOPHICAL LOGIC 209

55.4 Philosophical logic

Authors in philosophical logic often use deviating designations. A reflexive and a quasi-reflexive relation in themathematical sense is called a totally reflexive and a reflexive relation in philosophical logic sense, respectively.[4][5]

55.5 See also

• Binary relation

• Symmetric relation

• Transitive relation




210 CHAPTER 55. REFLEXIVE RELATION

• Coreflexive relation

55.6 Notes[1] Levy 1979:74

[2] Relational Mathematics, 2010

[3] On-Line Encyclopedia of Integer Sequences A053763

[4] Alan Hausman, Howard Kahane, Paul Tidman (2013). Logic and Philosophy—AModern Introduction. Wadsworth. ISBN1-133-05000-X. Here: p.327-328

[5] D.S. Clarke, Richard Behling (1998). Deductive Logic — An Introduction to Evaluation Techniques and Logical Theory.University Press of America. ISBN 0-7618-0922-8. Here: p.187

55.7 References• Levy, A. (1979) Basic Set Theory, Perspectives in Mathematical Logic, Springer-Verlag. Reprinted 2002,

Dover. ISBN 0-486-42079-5

• Lidl, R. and Pilz, G. (1998). Applied abstract algebra, Undergraduate Texts in Mathematics, Springer-Verlag.ISBN 0-387-98290-6

• Quine, W. V. (1951). Mathematical Logic, Revised Edition. Reprinted 2003, Harvard University Press. ISBN0-674-55451-5


55.8 External links• Hazewinkel, Michiel, ed. (2001), “Reflexivity”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-

010-4








https://en.wikipedia.org/wiki/Undergraduate_Texts_in_Mathematics





http://www.encyclopediaofmath.org/index.php?title=p/r080590






Chapter 56

Relation algebra

Not to be confused with relational algebra, a framework for finitary relations and relational databases.

In mathematics and abstract algebra, a relation algebra is a residuated Boolean algebra expanded with an involutioncalled converse, a unary operation. The motivating example of a relation algebra is the algebra 2X² of all binaryrelations on a set X, that is, subsets of the cartesian square X2, with R•S interpreted as the usual composition ofbinary relations R and S, and with the converse of R interpreted as the inverse relation.Relation algebra emerged in the 19th-century work of Augustus De Morgan and Charles Peirce, which culminatedin the algebraic logic of Ernst Schröder. The equational form of relation algebra treated here was developed byAlfred Tarski and his students, starting in the 1940s. Tarski and Givant (1987) applied relation algebra to a variable-free treatment of axiomatic set theory, with the implication that mathematics founded on set theory could itself beconducted without variables.

56.1 Definition

A relation algebra (L, ∧, ∨, −, 0, 1, •, I, ˘ ) is an algebraic structure equipped with the Boolean operations ofconjunction x∧y, disjunction x∨y, and negation x−, the Boolean constants 0 and 1, the relational operations of com-position x•y and converse x ˘ , and the relational constant I, such that these operations and constants satisfy certainequations constituting an axiomatization of relation algebras. A relation algebra is to a system of binary relationson a set containing the empty (0), complete (1), and identity (I) relations and closed under these five operations as agroup is to a system of permutations of a set containing the identity permutation and closed under composition andinverse.Following Jónsson and Tsinakis (1993) it is convenient to define additional operations x◁y = x•y˘ , and, dually, x▷y= x ˘ •y . Jónsson and Tsinakis showed that I◁x = x▷I, and that both were equal to x ˘ . Hence a relation algebra canequally well be defined as an algebraic structure (L, ∧, ∨, −, 0, 1, •, I, ◁, ▷). The advantage of this signature overthe usual one is that a relation algebra can then be defined in full simply as a residuated Boolean algebra for whichI◁x is an involution, that is, I◁(I◁x) = x . The latter condition can be thought of as the relational counterpart of theequation 1/(1/x) = x for ordinary arithmetic reciprocal, and some authors use reciprocal as a synonym for converse.Since residuated Boolean algebras are axiomatized with finitely many identities, so are relation algebras. Hence thelatter form a variety, the variety RA of relation algebras. Expanding the above definition as equations yields thefollowing finite axiomatization.

56.1.1 Axioms

The axioms B1-B10 below are adapted from Givant (2006: 283), and were first set out by Tarski in 1948.[1]

L is a Boolean algebra under binary disjunction, ∨, and unary complementation ()−:

B1: A ∨ B = B ∨ A

B2: A ∨ (B ∨ C) = (A ∨ B) ∨ C

211



https://en.wikipedia.org/wiki/Relational_database



https://en.wikipedia.org/wiki/Residuated_Boolean_algebra

https://en.wikipedia.org/wiki/Reduct










https://en.wikipedia.org/wiki/Algebraic_logic




https://en.wikipedia.org/wiki/Introduction_to_Boolean_algebra



https://en.wikipedia.org/wiki/Signature_(logic)

https://en.wikipedia.org/wiki/Residuated_Boolean_algebra

https://en.wikipedia.org/wiki/Multiplicative_inverse

https://en.wikipedia.org/wiki/Variety_(universal_algebra)



https://en.wikipedia.org/wiki/Disjunction

https://en.wikipedia.org/wiki/Complement_(order_theory)

212 CHAPTER 56. RELATION ALGEBRA

B3: (A− ∨ B)− ∨ (A− ∨ B−)− = A

This axiomatization of Boolean algebra is due to Huntington (1933). Note that the meet of the implied Booleanalgebra is not the • operator (even though it distributes over ∨ like a meet does), nor is the 1 of the Boolean algebrathe I constant.L is a monoid under binary composition (•) and nullary identity I:

B4: A•(B•C) = (A•B)•CB5: A•I = A

Unary converse () ˘ is an involution with respect to composition:

B6: A ˘̆ = A

B7: (A•B) ˘ = B ˘ •A ˘

Axiom B6 defines conversion as an involution, whereas B7 expresses the antidistributive property of conversionrelative to composition.[2]

Converse and composition distribute over disjunction:

B8: (A∨B) ˘ = A ˘ ∨B ˘

B9: (A∨B)•C = (A•C)∨(B•C)

B10 is Tarski’s equational form of the fact, discovered by Augustus De Morgan, that A•B ≤ C− ↔ A ˘ •C ≤ B− ↔C•B ˘ ≤ A−.

B10: (A ˘ •(A•B)−)∨B− = B−

These axioms are ZFC theorems; for the purely Boolean B1-B3, this fact is trivial. After each of the following axiomsis shown the number of the corresponding theorem in chpt. 3 of Suppes (1960), an exposition of ZFC: B4 27, B545, B6 14, B7 26, B8 16, B9 23.

56.2 Expressing properties of binary relations in RA

The following table shows how many of the usual properties of binary relations can be expressed as succinct RAequalities or inequalities. Below, an inequality of the form A≤B is shorthand for the Boolean equation A∨B = B.The most complete set of results of this nature is chpt. C of Carnap (1958), where the notation is rather distant fromthat of this entry. Chpt. 3.2 of Suppes (1960) contains fewer results, presented as ZFC theorems and using a notationthat more resembles that of this entry. Neither Carnap nor Suppes formulated their results using the RA of this entry,or in an equational manner.

56.3 Expressive power

The metamathematics of RA are discussed at length in Tarski and Givant (1987), and more briefly in Givant (2006).RA consists entirely of equations manipulated using nothing more than uniform replacement and the substitution ofequals for equals. Both rules are wholly familiar from school mathematics and from abstract algebra generally. HenceRA proofs are carried out in a manner familiar to all mathematicians, unlike the case in mathematical logic generally.RA can express any (and up to logical equivalence, exactly the) first-order logic (FOL) formulas containing no morethan three variables. (A given variable can be quantified multiple times and hence quantifiers can be nested arbitrarilydeeply by “reusing” variables.) Surprisingly, this fragment of FOL suffices to express Peano arithmetic and almostall axiomatic set theories ever proposed. Hence RA is, in effect, a way of algebraizing nearly all mathematics,

https://en.wikipedia.org/wiki/Edward_Vermilye_Huntington



https://en.wikipedia.org/wiki/Nullary




https://en.wikipedia.org/wiki/Antidistributive

https://en.wikipedia.org/wiki/Distributive_law


https://en.wikipedia.org/wiki/ZFC



https://en.wikipedia.org/wiki/Metamathematics





https://en.wikipedia.org/wiki/Peano_arithmetic


56.4. EXAMPLES 213

while dispensing with FOL and its connectives, quantifiers, turnstiles, and modus ponens. Because RA can expressPeano arithmetic and set theory, Gödel’s incompleteness theorems apply to it; RA is incomplete, incompletable, andundecidable. (N.B. The Boolean algebra fragment of RA is complete and decidable.)The representable relation algebras, forming the class RRA, are those relation algebras isomorphic to some re-lation algebra consisting of binary relations on some set, and closed under the intended interpretation of the RAoperations. It is easily shown, e.g. using the method of pseudoelementary classes, that RRA is a quasivariety, thatis, axiomatizable by a universal Horn theory. In 1950, Roger Lyndon proved the existence of equations holding inRRA that did not hold in RA. Hence the variety generated by RRA is a proper subvariety of the variety RA. In1955, Alfred Tarski showed that RRA is itself a variety. In 1964, Donald Monk showed that RRA has no finiteaxiomatization, unlike RA which is finitely axiomatized by definition.

56.3.1 Q-Relation Algebras

An RA is a Q-Relation Algebra (QRA) if, in addition to B1-B10, there exist some A and B such that (Tarski andGivant 1987: §8.4):

Q0: A ˘ •A ≤ I

Q1: B ˘ •B ≤ I

Q2: A ˘ •B = 1

Essentially these axioms imply that the universe has a (non-surjective) pairing relation whose projections are A andB. It is a theorem that every QRA is a RRA (Proof by Maddux, see Tarski & Givant 1987: 8.4(iii) ).Every QRA is representable (Tarski and Givant 1987). That not every relation algebra is representable is a fun-damental way RA differs from QRA and Boolean algebras which, by Stone’s representation theorem for Booleanalgebras, are always representable as sets of subsets of some set, closed under union, intersection, and complement.

56.4 Examples

1. Any Boolean algebra can be turned into a RA by interpreting conjunction as composition (the monoid multipli-cation •), i.e. x•y is defined as x∧y. This interpretation requires that converse interpret identity (ў = y), and that bothresiduals y\x and x/y interpret the conditional y→x (i.e., ¬y∨x).2. The motivating example of a relation algebra depends on the definition of a binary relation R on a set X as anysubset R ⊆ X², where X² is the Cartesian square of X. The power set 2X² consisting of all binary relations on X isa Boolean algebra. While 2X² can be made a relation algebra by taking R•S = R∧S, as per example (1) above, thestandard interpretation of • is instead x(R•S)z = ∃y.xRySz. That is, the ordered pair (x,z) belongs to the relation R•Sjust when there exists y ∈ X such that (x,y) ∈ R and (y,z) ∈ S. This interpretation uniquely determines R\S as consistingof all pairs (y,z) such that for all x ∈ X, if xRy then xSz. Dually, S/R consists of all pairs (x,y) such that for all z ∈ X, ifyRz then xSz. The translation ў = ¬(y\¬I) then establishes the converse R ˘ of R as consisting of all pairs (y,x) suchthat (x,y) ∈ R.3. An important generalization of the previous example is the power set 2E where E ⊆ X² is any equivalence relation onthe set X. This is a generalization because X² is itself an equivalence relation, namely the complete relation consistingof all pairs. While 2E is not a subalgebra of 2X² when E ≠ X² (since in that case it does not contain the relation X²,the top element 1 being E instead of X²), it is nevertheless turned into a relation algebra using the same definitionsof the operations. Its importance resides in the definition of a representable relation algebra as any relation algebraisomorphic to a subalgebra of the relation algebra 2E for some equivalence relation E on some set. The previoussection says more about the relevant metamathematics.4. If group sum or product interprets composition, group inverse interprets converse, group identity interprets I, andif R is a one to one correspondence, so that R ˘ •R = R•R

˘ = I,[3] then L is a group as well as a monoid. B4-B7become well-known theorems of group theory, so that RA becomes a proper extension of group theory as well as ofBoolean algebra.

https://en.wikipedia.org/wiki/Logical_connective

https://en.wikipedia.org/wiki/Quantifier_(logic)

https://en.wikipedia.org/wiki/Turnstile_(symbol)

https://en.wikipedia.org/wiki/Modus_ponens

https://en.wikipedia.org/wiki/G%C3%B6del%2527s_incompleteness_theorems

https://en.wikipedia.org/wiki/G%C3%B6del%2527s_incompleteness_theorems

https://en.wikipedia.org/wiki/Undecidable_problem

https://en.wikipedia.org/wiki/Pseudoelementary_class

https://en.wikipedia.org/wiki/Quasivariety

https://en.wikipedia.org/wiki/Universal_Horn_theory

https://en.wikipedia.org/wiki/Roger_Lyndon



https://en.wikipedia.org/wiki/Stone%2527s_representation_theorem_for_Boolean_algebras

https://en.wikipedia.org/wiki/Stone%2527s_representation_theorem_for_Boolean_algebras




https://en.wikipedia.org/wiki/Group_inverse

https://en.wikipedia.org/wiki/One_to_one_correspondence




https://en.wikipedia.org/wiki/Proper_extension


214 CHAPTER 56. RELATION ALGEBRA

56.5 Historical remarks

DeMorgan founded RA in 1860, but C. S. Peirce took it much further and became fascinated with its philosophicalpower. The work of DeMorgan and Peirce came to be known mainly in the extended and definitive form ErnstSchröder gave it in Vol. 3 of his Vorlesungen (1890–1905). Principia Mathematica drew strongly on Schröder’s RA,but acknowledged him only as the inventor of the notation. In 1912, Alwin Korselt proved that a particular formulain which the quantifiers were nested four deep had no RA equivalent.[4] This fact led to a loss of interest in RA untilTarski (1941) began writing about it. His students have continued to develop RA down to the present day. Tarskireturned to RA in the 1970s with the help of Steven Givant; this collaboration resulted in the monograph by Tarskiand Givant (1987), the definitive reference for this subject. For more on the history ofRA, see Maddux (1991, 2006).

56.6 Software

• RelMICS / Relational Methods in Computer Science maintained by Wolfram Kahl

• Carsten Sinz: ARA / An Automatic Theorem Prover for Relation Algebras

56.7 See also

56.8 Footnotes[1] Alfred Tarski (1948) “Abstract: Representation Problems for Relation Algebras,” Bulletin of the AMS 54: 80.

[2] Chris Brink; Wolfram Kahl; Gunther Schmidt (1997). Relational Methods in Computer Science. Springer. pp. 4 and 8.ISBN 978-3-211-82971-4.

[3] Tarski, A. (1941), p. 87.

[4] Korselt did not publish his finding. It was first published in Leopold Loewenheim (1915) "Über Möglichkeiten im Rela-tivkalkül,” Mathematische Annalen 76: 447–470. Translated as “On possibilities in the calculus of relatives” in Jean vanHeijenoort, 1967. A Source Book in Mathematical Logic, 1879–1931. Harvard Univ. Press: 228–251.

56.9 References

• Rudolf Carnap (1958) Introduction to Symbolic Logic and its Applications. Dover Publications.

• Givant, Steven (2006). “The calculus of relations as a foundation for mathematics”. Journal of AutomatedReasoning 37: 277–322. doi:10.1007/s10817-006-9062-x.

• Halmos, P. R., 1960. Naive Set Theory. Van Nostrand.

• Leon Henkin, Alfred Tarski, and Monk, J. D., 1971. Cylindric Algebras, Part 1, and 1985, Part 2. NorthHolland.

• Hirsch R., and Hodkinson, I., 2002, Relation Algebra byGames, vol. 147 in Studies in Logic and the Foundationsof Mathematics. Elsevier Science.

• Jónsson, Bjarni; Tsinakis, Constantine (1993). “Relation algebras as residuated Boolean algebras”. AlgebraUniversalis 30: 469–78. doi:10.1007/BF01195378.

• Maddux, Roger (1991). “The Origin of Relation Algebras in the Development and Axiomatization of theCalculus of Relations” (PDF). Studia Logica 50 (3–4): 421–455. doi:10.1007/BF00370681.

• --------, 2006. Relation Algebras, vol. 150 in Studies in Logic and the Foundations of Mathematics. ElsevierScience.

• Patrick Suppes, 1960. Axiomatic Set Theory. Van Nostrand. Dover reprint, 1972. Chpt. 3.

https://en.wikipedia.org/wiki/DeMorgan





https://en.wikipedia.org/wiki/Alwin_Korselt

http://relmics.mcmaster.ca/html/index.html

http://www.cas.mcmaster.ca/~kahl/

http://www-sr.informatik.uni-tuebingen.de/~sinz/ARA/





https://en.wikipedia.org/wiki/Leopold_Loewenheim

https://en.wikipedia.org/wiki/Mathematische_Annalen

https://en.wikipedia.org/wiki/Jean_van_Heijenoort

https://en.wikipedia.org/wiki/Jean_van_Heijenoort

https://en.wikipedia.org/wiki/Rudolf_Carnap


https://dx.doi.org/10.1007%252Fs10817-006-9062-x

https://en.wikipedia.org/wiki/Paul_Richard_Halmos

https://en.wikipedia.org/wiki/Leon_Henkin


http://www.elsevier.com/wps/find/bookdescription.cws_home/625473/description#description

https://en.wikipedia.org/wiki/Bjarni_J%C3%B3nsson


https://dx.doi.org/10.1007%252FBF01195378

https://en.wikipedia.org/wiki/Roger_Maddux

http://orion.math.iastate.edu/maddux/papers/Maddux1991.pdf

http://orion.math.iastate.edu/maddux/papers/Maddux1991.pdf





• Gunther Schmidt, 2010. Relational Mathematics. Cambridge University Press.

• Tarski, Alfred (1941). “On the calculus of relations”. Journal of Symbolic Logic 6: 73–89. doi:10.2307/2268577.

• ------, and Givant, Steven, 1987. A Formalization of Set Theory without Variables. Providence RI: AmericanMathematical Society.

56.10 External links• Yohji AKAMA, Yasuo Kawahara, and Hitoshi Furusawa, "Constructing Allegory from Relation Algebra and

Representation Theorems."

• Richard Bird, Oege de Moor, Paul Hoogendijk, "Generic Programming with Relations and Functors."

• R.P. de Freitas and Viana, "A Completeness Result for Relation Algebra with Binders."

• Peter Jipsen:

• Relation algebras. In Mathematical structures. If there are problems with LaTeX, see an old HTMLversion here.

• "Foundations of Relations and Kleene Algebra."• "Computer Aided Investigations of Relation Algebras."• "A Gentzen System And Decidability For Residuated Lattices.”

• Vaughan Pratt:

• "Origins of the Calculus of Binary Relations." A historical treatment.• "The Second Calculus of Binary Relations."

• Priss, Uta:

• "An FCA interpretation of Relation Algebra."• "Relation Algebra and FCA" Links to publications and software

• Kahl, Wolfram, and Schmidt, Gunther, "Exploring (Finite) Relation Algebras Using Tools Written in Haskell."See homepage of the whole project.



http://www.jstor.org/stable/2268577


https://dx.doi.org/10.2307%252F2268577

http://nicosia.is.s.u-tokyo.ac.jp/pub/staff/akama/repr.ps

http://nicosia.is.s.u-tokyo.ac.jp/pub/staff/akama/repr.ps

http://citeseer.ist.psu.edu/bird99generic.html

http://www.cos.ufrj.br/~naborges/fv02.ps

http://www1.chapman.edu/~jipsen/

http://math.chapman.edu/structuresold/files/Relation_algebras.pdf

http://math.chapman.edu/cgi-bin/structures

http://math.chapman.edu/cgi-bin/structures.pl?Relation_algebras

http://math.chapman.edu/~jipsen/talks/RelMiCS2006/JipsenRAKAtutorial.pdf

http://www1.chapman.edu/~jipsen/dissertation/

http://citeseer.ist.psu.edu/337149.html

https://en.wikipedia.org/wiki/Vaughan_Pratt

http://boole.stanford.edu/pub/ocbr.pdf

http://boole.stanford.edu/pub/scbr.pdf

http://www.upriss.org.uk/papers/fcaic06.pdf

http://www.upriss.org.uk/fca/relalg.html

http://www.cas.mcmaster.ca/~kahl/

http://ist.unibw-muenchen.de/People/schmidt/

http://relmics.mcmaster.ca/~kahl/Publications/TR/2000-02/

http://relmics.mcmaster.ca/tools/RATH/index.html

Chapter 57

Relation construction

In logic and mathematics, relation construction and relational constructibility have to do with the ways that onerelation is determined by an indexed family or a sequence of other relations, called the relation dataset. The relationin the focus of consideration is called the faciendum. The relation dataset typically consists of a specified relationover sets of relations, called the constructor, the factor, or the method of construction, plus a specified set of otherrelations, called the faciens, the ingredients, or the makings.Relation composition and relation reduction are special cases of relation constructions.

57.1 See also• Projection

• Relation

• Relation composition

• Relation reduction

216




https://en.wikipedia.org/wiki/Indexed_family



https://en.wikipedia.org/wiki/Relation_reduction

https://en.wikipedia.org/wiki/Projection_(mathematics)



https://en.wikipedia.org/wiki/Relation_reduction

Chapter 58

Representation (mathematics)

In mathematics, representation is a very general relationship that expresses similarities between objects. Roughlyspeaking, a collection Y of mathematical objects may be said to represent another collection X of objects, providedthat the properties and relationships existing among the representing objects yi conform in some consistent way tothose existing among the corresponding represented objects xi. Somewhat more formally, for a set Π of propertiesand relations, a Π-representation of some structure X is a structure Y that is the image of X under a homomorphismthat preserves Π. The label representation is sometimes also applied to the homomorphism itself.

58.1 Representation theory

Perhaps the most well-developed example of this general notion is the subfield of abstract algebra called representationtheory, which studies the representing of elements of algebraic structures by linear transformations of vector spaces.

58.2 Other examples

Although the term representation theory is well established in the algebraic sense discussed above, there are manyother uses of the term representation throughout mathematics.

58.2.1 Graph theory

An active area of graph theory is the exploration of isomorphisms between graphs and other structures. A keyclass of such problems stems from the fact that, like adjacency in undirected graphs, intersection of sets (or, moreprecisely, non-disjointness) is a symmetric relation. This gives rise to the study of intersection graphs for innumerablefamilies of sets.[1] One foundational result here, due to Paul Erdős and colleagues, is that every n-vertex graph maybe represented in terms of intersection among subsets of a set of size no more than n2/4.[2]

Representing a graph by such algebraic structures as its adjacency matrix and Laplacian matrix gives rise to the fieldof spectral graph theory.[3]

58.2.2 Order theory

Dual to the observation above that every graph is an intersection graph is the fact that every partially ordered set isisomorphic to a collection of sets ordered by the containment (or inclusion) relation ⊆. Among the posets that ariseas the containment orders for natural classes of objects are the Boolean lattices and the orders of dimension n.[4]

Many partial orders arise from (and thus can be represented by) collections of geometric objects. Among them arethe n-ball orders. The 1-ball orders are the interval-containment orders, and the 2-ball orders are the so-called circleorders, the posets representable in terms of containment among disks in the plane. A particularly nice result in thisfield is the characterization of the planar graphs as those graphs whose vertex-edge incidence relations are circleorders.[5]

217





https://en.wikipedia.org/wiki/Representation_theory

https://en.wikipedia.org/wiki/Representation_theory

https://en.wikipedia.org/wiki/Algebraic_structures

https://en.wikipedia.org/wiki/Linear_transformations

https://en.wikipedia.org/wiki/Vector_spaces



https://en.wikipedia.org/wiki/Adjacency_relation

https://en.wikipedia.org/wiki/Undirected_graph

https://en.wikipedia.org/wiki/Intersection_(mathematics)



https://en.wikipedia.org/wiki/Intersection_graph

https://en.wikipedia.org/wiki/Paul_Erd%C5%91s

https://en.wikipedia.org/wiki/Vertex_(graph_theory)

https://en.wikipedia.org/wiki/Subsets

https://en.wikipedia.org/wiki/Adjacency_matrix

https://en.wikipedia.org/wiki/Laplacian_matrix

https://en.wikipedia.org/wiki/Spectral_graph_theory

https://en.wikipedia.org/wiki/Dual_(mathematics)



https://en.wikipedia.org/wiki/Containment_order

https://en.wikipedia.org/wiki/Boolean_lattice

https://en.wikipedia.org/wiki/Order_dimension


https://en.wikipedia.org/wiki/N-sphere#n-ball

https://en.wikipedia.org/wiki/Planar_graph

218 CHAPTER 58. REPRESENTATION (MATHEMATICS)

There are also geometric representations that are not based on containment. Indeed, one of the best studied classesamong these are the interval orders,[6] which represent the partial order in terms of what might be called disjointprecedence of intervals on the real line: each element x of the poset is represented by an interval [x1, x2] such that forany y and z in the poset, y is below z if and only if y2 < z1.

58.2.3 Polysemy

Under certain circumstances, a single function f:X → Y is at once an isomorphism from several mathematical struc-tures on X. Since each of those structures may be thought of, intuitively, as a meaning of the image Y—one of thethings that Y is trying to tell us—this phenomenon is called polysemy, a term borrowed from linguistics. Examplesinclude:

• intersection polysemy—pairs of graphs G1 and G2 on a common vertex set V that can be simultaneouslyrepresented by a single collection of sets Sv such that any distinct vertices u and w in V...

are adjacent in G1 if and only if their corresponding sets intersect ( Su ∩ Sw ≠ Ø ), andare adjacent in G2 if and only if the complements do ( SuC ∩ SwC ≠ Ø ).[7]

• competition polysemy—motivated by the study of ecological food webs, in which pairs of species may haveprey in common or have predators in common. A pair of graphs G1 and G2 on one vertex set is competitionpolysemic if and only if there exists a single directed graph D on the same vertex set such that any distinctvertices u and v...

are adjacent in G1 if and only if there is a vertex w such that both uw and vw are arcs in D,andare adjacent in G2 if and only if there is a vertex w such that both wu and wv are arcs in D.[8]

• interval polysemy—pairs of posets P1 and P2 on a common ground set that can be simultaneously representedby a single collection of real intervals that is an interval-order representation of P1 and an interval-containmentrepresentation of P2.[9]

58.3 See also• Representation theorems

• Model theory

58.4 References[1] • McKee, Terry A.; McMorris, F. R. (1999), Topics in Intersection Graph Theory, SIAM Monographs on Discrete

Mathematics and Applications, Philadelphia: Society for Industrial and Applied Mathematics, ISBN 0-89871-430-3, MR 1672910

[2] Erdős, Paul; Goodman, A. W.; Pósa, Louis (1966), “The representation of a graph by set intersections”, Canadian Journalof Mathematics 18 (1): 106–112, doi:10.4153/cjm-1966-014-3, MR 0186575

[3] • Biggs, Norman (1994), Algebraic Graph Theory, Cambridge Mathematical Library, Cambridge University Press,ISBN 978-0-521-45897-9, MR 1271140

[4] • Trotter, William T. (1992), Combinatorics and Partially Ordered Sets: Dimension Theory, Johns Hopkins Series in theMathematical Sciences, Baltimore: The Johns Hopkins University Press, ISBN 978-0-8018-4425-6, MR 1169299

[5] • Scheinerman, Edward (1991), “A note on planar graphs and circle orders”, SIAM Journal on Discrete Mathematics 4(3): 448–451, doi:10.1137/0404040, MR 1105950

[6] • Fishburn, Peter C. (1985), Interval Orders and Interval Graphs: A Study of Partially Ordered Sets, Wiley-InterscienceSeries in Discrete Mathematics, John Wiley & Sons, ISBN 978-0-471-81284-5, MR 0776781


https://en.wikipedia.org/wiki/Real_line

https://en.wikipedia.org/wiki/Polysemy

https://en.wikipedia.org/wiki/Set_complement

https://en.wikipedia.org/wiki/Ecology

https://en.wikipedia.org/wiki/Food_web


https://en.wikipedia.org/wiki/Arc_(graph_theory)

https://en.wikipedia.org/wiki/Representation_theorem







https://en.wikipedia.org/wiki/Paul_Erd%C5%91s

https://en.wikipedia.org/wiki/Lajos_P%C3%B3sa_(mathematician)

https://en.wikipedia.org/wiki/Canadian_Journal_of_Mathematics

https://en.wikipedia.org/wiki/Canadian_Journal_of_Mathematics


https://dx.doi.org/10.4153%252Fcjm-1966-014-3











https://en.wikipedia.org/wiki/Ed_Scheinerman

https://en.wikipedia.org/wiki/SIAM_Journal_on_Discrete_Mathematics


https://dx.doi.org/10.1137%252F0404040








[7] • Tanenbaum, Paul J. (1999), “Simultaneous intersection representation of pairs of graphs”, Journal of Graph Theory32 (2): 171–190, doi:10.1002/(SICI)1097-0118(199910)32:2<171::AID-JGT7>3.0.CO;2-N, MR 1709659

[8] • Fischermann, Miranca; Knoben, Werner; Kremer, Dirk; Rautenbachh, Dieter (2004), “Competition polysemy”,Discrete Mathematics 282 (1–3): 251–255, doi:10.1016/j.disc.2003.11.014, MR 2059526

[9] • Tanenbaum, Paul J. (1996), “Simultaneous representation of interval and interval-containment orders”, Order 13 (4):339–350, doi:10.1007/BF00405593, MR 1452517

https://en.wikipedia.org/wiki/Journal_of_Graph_Theory


https://dx.doi.org/10.1002%252F%2528SICI%25291097-0118%2528199910%252932%253A2%253C171%253A%253AAID-JGT7%253E3.0.CO%253B2-N



https://en.wikipedia.org/wiki/Discrete_Mathematics_(journal)


https://dx.doi.org/10.1016%252Fj.disc.2003.11.014



https://en.wikipedia.org/wiki/Order_(journal)





Chapter 59

Semiorder

In order theory, a branch of mathematics, a semiorder is a type of ordering that may be determined for a set ofitems with numerical scores by declaring two items to be incomparable when their scores are within a given marginof error of each other, and by using the numerical comparison of their scores when those scores are sufficiently farapart. Semiorders were introduced and applied in mathematical psychology by Luce (1956) as a model of humanpreference without the assumption that indifference is transitive. They generalize strict weak orderings, form a specialcase of partial orders and interval orders, and can be characterized among the partial orders by two forbidden four-item suborders.

59.1 Definition

Let X be a set of items, and let < be a binary relation on X. Items x and y are said to be incomparable, written here asx ~ y, if neither x < y nor y < x is true. Then the pair (X,<) is a semiorder if it satisfies the following three axioms:[1]

• For all x and y, it is not possible for both x < y and y < x to be true. That is, < must be an irreflexive,antisymmetric relation

• For all x, y, z, and w, if it is true that x < y, y ~ z, and z < w, then it must also be true that x < w.

• For all x, y, z, and w, if it is true that x < y, y < z, and y ~ w, then it cannot also be true that x ~ w and z ~ wsimultaneously.

It follows from the first axiom that x ~ x, and therefore the second axiom (with y = z) implies that < is a transitiverelation.One may define a partial order (X,≤) from a semiorder (X,<) by declaring that x ≤ y whenever either x < y or x = y. Ofthe axioms that a partial order is required to obey, reflexivity follows automatically from this definition, antisymmetryfollows from the first semiorder axiom, and transitivity follows from the second semiorder axiom. Conversely, froma partial order defined in this way, the semiorder may be recovered by declaring that x < y whenever x ≤ y and x ≠y. The first of the semiorder axioms listed above follows automatically from the axioms defining a partial order, butthe others do not. The second and third semiorder axioms forbid partial orders of four items forming two disjointchains: the second axiom forbids two chains of two items each, while the third item forbids a three-item chain withone unrelated item.

59.2 Utility

The original motivation for introducing semiorders was to model human preferences without assuming (as strict weakorderings do) that incomparability is a transitive relation. For instance, if x, y, and z represent three quantities of thesame material, and x and z differ by the smallest amount that is perceptible as a difference, while y is halfway betweenthe two of them, then it is reasonable for a preference to exist between x and z but not between the other two pairs,violating transitivity.[2]

220


https://en.wikipedia.org/wiki/Margin_of_error


https://en.wikipedia.org/wiki/Mathematical_psychology

https://en.wikipedia.org/wiki/Semiorder#CITEREFLuce1956












59.2. UTILITY 221

An example of a semiorder, shown by its Hasse diagram. The horizontal blue lines indicate the spacing of the y-coordinates of thepoints; two points are comparable when their y coordinates differ by at least one unit.

Thus, suppose that X is a set of items, and u is a utility function that maps the members of X to real numbers. A strictweak ordering can be defined on x by declaring two items to be incomparable when they have equal utilities, andotherwise using the numerical comparison, but this necessarily leads to a transitive incomparability relation. Instead,if one sets a numerical threshold (which may be normalized to 1) such that utilities within that threshold of each otherare declared incomparable, then a semiorder arises.Specifically, define a binary relation < from X and u by setting x < y whenever u(x) ≤ u(y) − 1. Then (X,<) is asemiorder.[3] It may equivalently be defined as the interval order defined by the intervals [u(x),u(x) + 1].[4]

The converse is not necessarily true: for instance, if a semiorder (X,<) includes an uncountable totally ordered subsetthen there do not exist sufficiently many sufficiently well-spaced real-numbers to represent this subset numerically.However, every finite semiorder can be defined from a utility function in this way.[5] Fishburn (1973) supplies aprecise characterization of the semiorders that may be defined numerically.


https://en.wikipedia.org/wiki/Utility_function



https://en.wikipedia.org/wiki/Uncountable


https://en.wikipedia.org/wiki/Semiorder#CITEREFFishburn1973

222 CHAPTER 59. SEMIORDER

59.3 Other results

The number of distinct semiorders on n unlabeled items is given by the Catalan numbers

1n+1

(2nn

), [6]

while the number of semiorders on n labeled items is given by the sequence

1, 1, 3, 19, 183, 2371, 38703, 763099, 17648823, ... (sequence A006531 in OEIS).[7]

Any finite semiorder has order dimension at most three.[8]

Among all partial orders with a fixed number of elements and a fixed number of comparable pairs, the partial ordersthat have the largest number of linear extensions are semiorders.[9]

Semiorders are known to obey the 1/3–2/3 conjecture: in any finite semiorder that is not a total order, there existsa pair of elements x and y such that x appears earlier than y in between 1/3 and 2/3 of the linear extensions of thesemiorder.[10]

The set of semiorders on an n-element set is well-graded: if two semiorders on the same set differ from each otherby the addition or removal of k order relations, then it is possible to find a path of k steps from the first semiorder tothe second one, in such a way that each step of the path adds or removes a single order relation and each intermediatestate in the path is itself a semiorder.[11]

The incomparability graphs of semiorders are called indifference graphs, and are a special case of the intervalgraphs.[12]

59.4 Notes

[1] Luce (1956) describes an equivalent set of four axioms, the first two of which combine the definition of incomparabilityand the first axiom listed here.

[2] Luce (1956), p. 179.

[3] Luce (1956), Theorem 3 describes a more general situation in which the threshold for comparability between two utilitiesis a function of the utility rather than being identically 1.

[4] Fishburn (1970).

[5] This result is typically credited to Scott & Suppes (1958); see, e.g., Rabinovitch (1977). However, Luce (1956), Theorem2 proves a more general statement, that a finite semiorder can be defined from a utility function and a threshold functionwhenever a certain underlying weak order can be defined numerically. For finite semiorders, it is trivial that the weak ordercan be defined numerically with a unit threshold function.

[6] Kim & Roush (1978).

[7] Chandon, Lemaire & Pouget (1978).

[8] Rabinovitch (1978).

[9] Fishburn & Trotter (1992).

[10] Brightwell (1989).

[11] Doignon & Falmagne (1997).

[12] Roberts, Fred S. (1969), “Indifference graphs”, Proof Techniques in Graph Theory (Proc. Second Ann Arbor Graph TheoryConf., Ann Arbor, Mich., 1968), Academic Press, New York, pp. 139–146, MR 0252267.

https://en.wikipedia.org/wiki/Catalan_number





https://en.wikipedia.org/wiki/1/3%E2%80%932/3_conjecture


https://en.wikipedia.org/wiki/Indifference_graph

https://en.wikipedia.org/wiki/Interval_graph





https://en.wikipedia.org/wiki/Semiorder#CITEREFFishburn1970

https://en.wikipedia.org/wiki/Semiorder#CITEREFScottSuppes1958

https://en.wikipedia.org/wiki/Semiorder#CITEREFRabinovitch1977


https://en.wikipedia.org/wiki/Semiorder#CITEREFKimRoush1978

https://en.wikipedia.org/wiki/Semiorder#CITEREFChandonLemairePouget1978

https://en.wikipedia.org/wiki/Semiorder#CITEREFRabinovitch1978

https://en.wikipedia.org/wiki/Semiorder#CITEREFFishburnTrotter1992

https://en.wikipedia.org/wiki/Semiorder#CITEREFBrightwell1989

https://en.wikipedia.org/wiki/Semiorder#CITEREFDoignonFalmagne1997

https://en.wikipedia.org/wiki/Fred_S._Roberts




59.5 References• Brightwell, Graham R. (1989), “Semiorders and the 1/3–2/3 conjecture”, Order 5 (4): 369–380, doi:10.1007/BF00353656.

• Chandon, J.-L.; Lemaire, J.; Pouget, J. (1978), “Dénombrement des quasi-ordres sur un ensemble fini”, Centrede Mathématique Sociale. École Pratique des Hautes Études. Mathématiques et Sciences Humaines (62): 61–80,83, MR 517680.

• Doignon, Jean-Paul; Falmagne, Jean-Claude (1997), “Well-graded families of relations”, Discrete Mathematics173 (1-3): 35–44, doi:10.1016/S0012-365X(96)00095-7, MR 1468838.

• Fishburn, Peter C. (1970), “Intransitive indifference with unequal indifference intervals”, J. MathematicalPsychology 7: 144–149, doi:10.1016/0022-2496(70)90062-3, MR 0253942.

• Fishburn, Peter C. (1973), “Interval representations for interval orders and semiorders”, J. Mathematical Psy-chology 10: 91–105, doi:10.1016/0022-2496(73)90007-2, MR 0316322.

• Fishburn, Peter C.; Trotter, W. T. (1992), “Linear extensions of semiorders: a maximization problem”, DiscreteMathematics 103 (1): 25–40, doi:10.1016/0012-365X(92)90036-F, MR 1171114.

• Kim, K. H.; Roush, F. W. (1978), “Enumeration of isomorphism classes of semiorders”, Journal of Combina-torics, Information &System Sciences 3 (2): 58–61, MR 538212.

• Luce, R. Duncan (1956), “Semiorders and a theory of utility discrimination”, Econometrica 24: 178–191,JSTOR 1905751, MR 0078632.

• Rabinovitch, Issie (1977), “The Scott-Suppes theorem on semiorders”, J. Mathematical Psychology 15 (2):209–212, doi:10.1016/0022-2496(77)90030-x, MR 0437404.

• Rabinovitch, Issie (1978), “The dimension of semiorders”, Journal of Combinatorial Theory. Series A 25 (1):50–61, doi:10.1016/0097-3165(78)90030-4, MR 0498294.

• Scott, Dana; Suppes, Patrick (1958), “Foundational aspects of theories of measurement”, The Journal of Sym-bolic Logic 23: 113–128, doi:10.2307/2964389, MR 0115919.

59.6 Additional reading• Pirlot, M.; Vincke, Ph. (1997), Semiorders: Properties, representations, applications, Theory and Decision

Library. Series B: Mathematical and Statistical Methods 36, Dordrecht: Kluwer Academic Publishers Group,ISBN 0-7923-4617-3, MR 1472236.

https://en.wikipedia.org/wiki/Graham_Brightwell






https://en.wikipedia.org/wiki/Jean-Claude_Falmagne





https://en.wikipedia.org/wiki/Peter_C._Fishburn


https://dx.doi.org/10.1016%252F0022-2496%252870%252990062-3





https://dx.doi.org/10.1016%252F0022-2496%252873%252990007-2





https://dx.doi.org/10.1016%252F0012-365X%252892%252990036-F





https://en.wikipedia.org/wiki/R._Duncan_Luce






https://dx.doi.org/10.1016%252F0022-2496%252877%252990030-x




https://dx.doi.org/10.1016%252F0097-3165%252878%252990030-4



https://en.wikipedia.org/wiki/Dana_Scott



https://dx.doi.org/10.2307%252F2964389







Chapter 60

Separoid

In mathematics, a separoid is a binary relation between disjoint sets which is stable as an ideal in the canonical orderinduced by inclusion. Many mathematical objects which appear to be quite different, find a common generalisation inthe framework of separoids; e.g., graphs, configurations of convex sets, oriented matroids, and polytopes. Any count-able category is an induced subcategory of separoids when they are endowed with homomorphisms (viz., mappingsthat preserve the so-called minimal Radon partitions).In this general framework, some results and invariants of different categories turn out to be special cases of thesame aspect; e.g., the pseudoachromatic number from graph theory and the Tverberg theorem from combinatorialconvexity are simply two faces of the same aspect, namely, complete colouring of separoids.

60.1 The axioms

A separoid is a set S endowed with a binary relation | ⊆ 2S × 2S on its power set, which satisfies the followingsimple properties for A,B ⊆ S :

A | B ⇔ B | A,

A | B ⇒ A ∩B = ∅,

A | B and A′ ⊂ A⇒ A′ | B.

A related pair A | B is called a separation and we often say that A is separated from B. It is enough to know themaximal separations to reconstruct the separoid.A mapping φ : S → T is a morphism of separoids if the preimages of separations are separations; that is, forA,B ⊆ T

A | B ⇒ φ−1(A) | φ−1(B).

60.2 Examples

Examples of separoids can be found in almost every branch of mathematics. Here we list just a few.1. Given a graph G=(V,E), we can define a separoid on its vertices by saying that two (disjoint) subsets of V, say Aand B, are separated if there are no edges going from one to the other; i.e.,

A | B ⇔ ∀a ∈ A and b ∈ B : ab ̸∈ E.

224




https://en.wikipedia.org/wiki/Ideal_(order_theory)



https://en.wikipedia.org/wiki/Convex_set

https://en.wikipedia.org/wiki/Oriented_matroid

https://en.wikipedia.org/wiki/Polytopes



https://en.wikipedia.org/wiki/Radon%2527s_theorem

https://en.wikipedia.org/wiki/Tverberg_theorem







https://en.wikipedia.org/wiki/Vertex_(graph_theory)

https://en.wikipedia.org/wiki/Edge_(graph_theory)

60.3. THE BASIC LEMMA 225

2. Given an oriented matroid M = (E,T), given in terms of its topes T, we can define a separoid on E by saying thattwo subsets are separated if they are contained in opposite signs of a tope. In other words, the topes of an orientedmatroid are the maximal separations of a separoid. This example includes, of course, all directed graphs.3. Given a family of objects in an Euclidean space, we can define a separoid in it by saying that two subsets areseparated if there exists a hyperplane that separates them; i.e., leaving them in the two opposite sides of it.4. Given a topological space, we can define a separoid saying that two subsets are separated if there exist two disjointopen sets which contains them (one for each of them).

60.3 The basic lemma

Every separoid can be represented with a family of convex sets in some Euclidean space and their separations byhyperplanes.

60.4 References• Strausz Ricardo; “Separoides”. Situs, serie B, no. 5 (1998), Universidad Nacional Autónoma de México.

• Arocha Jorge Luis, Bracho Javier, Montejano Luis, Oliveros Deborah, Strausz Ricardo; “Separoids, their cat-egories and a Hadwiger-type theorem for transversals”. Discrete and Computational Geometry 27 (2002), no.3, 377–385.

• Strausz Ricardo; “Separoids and a Tverberg-type problem”. Geombinatorics 15 (2005), no. 2, 79–92.

• Montellano-Ballesteros Juan Jose, Por Attila, Strausz Ricardo; “Tverberg-type theorems for separoids”. Dis-crete and Computational Geometry 35 (2006), no.3, 513–523.

• Nešetřil Jaroslav, Strausz Ricardo; “Universality of separoids”. ArchivumMathematicum (Brno) 42 (2006), no.1, 85–101.

• Bracho Javier, Strausz Ricardo; “Two geometric representations of separoids”. Periodica Mathematica Hun-garica 53 (2006), no. 1-2, 115–120.

• Strausz Ricardo; “Homomorphisms of separoids”. 6th Czech-Slovak International Symposium on Combina-torics, Graph Theory, Algorithms and Applications, 461–468, Electronic Notes on Discrete Mathematics 28,Elsevier, Amsterdam, 2007.

• Strausz Ricardo; “Edrös-Szekeres 'happy end'-type theorems for separoids”. European Journal of Combina-torics 29 (2008), no. 4, 1076–1085.


https://en.wikipedia.org/wiki/Euclidean_space

https://en.wikipedia.org/wiki/Hyperplane


https://en.wikipedia.org/wiki/Open_set

https://en.wikipedia.org/wiki/Hadwiger_transversal_theorem

https://en.wikipedia.org/wiki/Discrete_and_Computational_Geometry

https://en.wikipedia.org/wiki/Geombinatorics

https://en.wikipedia.org/wiki/Jaroslav_Ne%C5%A1et%C5%99il

http://atenea.matem.unam.mx/EMIS/journals/AM/06-1/neset1.pdf

Chapter 61

Sequential composition

In computer science, the process calculi (or process algebras) are a diverse family of related approaches for for-mally modelling concurrent systems. Process calculi provide a tool for the high-level description of interactions,communications, and synchronizations between a collection of independent agents or processes. They also providealgebraic laws that allow process descriptions to be manipulated and analyzed, and permit formal reasoning aboutequivalences between processes (e.g., using bisimulation). Leading examples of process calculi include CSP, CCS,ACP, and LOTOS.[1] More recent additions to the family include the π-calculus, the ambient calculus, PEPA, thefusion calculus and the join-calculus.

61.1 Essential features

While the variety of existing process calculi is very large (including variants that incorporate stochastic behaviour,timing information, and specializations for studying molecular interactions), there are several features that all processcalculi have in common:[2]

• Representing interactions between independent processes as communication (message-passing), rather than asmodification of shared variables.

• Describing processes and systems using a small collection of primitives, and operators for combining thoseprimitives.

• Defining algebraic laws for the process operators, which allow process expressions to be manipulated usingequational reasoning.

61.2 Mathematics of processes

To define a process calculus, one starts with a set of names (or channels) whose purpose is to provide means ofcommunication. In many implementations, channels have rich internal structure to improve efficiency, but this isabstracted away in most theoretic models. In addition to names, one needs a means to form new processes from old.The basic operators, always present in some form or other, allow:[3]

• parallel composition of processes

• specification of which channels to use for sending and receiving data

• sequentialization of interactions

• hiding of interaction points

• recursion or process replication

226


https://en.wikipedia.org/wiki/Concurrent_system

https://en.wikipedia.org/wiki/Algebra





https://en.wikipedia.org/wiki/Language_Of_Temporal_Ordering_Specification

https://en.wikipedia.org/wiki/Pi-calculus


https://en.wikipedia.org/wiki/PEPA

https://en.wikipedia.org/wiki/Fusion_calculus

https://en.wikipedia.org/wiki/Join-calculus

https://en.wikipedia.org/wiki/Stochastic

https://en.wikipedia.org/wiki/Message-passing

https://en.wikipedia.org/wiki/Equational_reasoning

https://en.wikipedia.org/wiki/Channel_(programming)

61.2. MATHEMATICS OF PROCESSES 227

61.2.1 Parallel composition

Parallel composition of two processes P and Q , usually written P |Q , is the key primitive distinguishing the processcalculi from sequential models of computation. Parallel composition allows computation in P and Q to proceedsimultaneously and independently. But it also allows interaction, that is synchronisation and flow of information fromP to Q (or vice versa) on a channel shared by both. Crucially, an agent or process can be connected to more than onechannel at a time.Channels may be synchronous or asynchronous. In the case of a synchronous channel, the agent sending a messagewaits until another agent has received the message. Asynchronous channels do not require any such synchronization.In some process calculi (notably the π-calculus) channels themselves can be sent in messages through (other) channels,allowing the topology of process interconnections to change. Some process calculi also allow channels to be createdduring the execution of a computation.

61.2.2 Communication

Interaction can be (but isn't always) a directed flow of information. That is, input and output can be distinguished asdual interaction primitives. Process calculi that make such distinctions typically define an input operator (e.g. x(v)) and an output operator (e.g. x⟨y⟩ ), both of which name an interaction point (here x ) that is used to synchronisewith a dual interaction primitive.Information should be exchanged, it will flow from the outputting to the inputting process. The output primitive willspecify the data to be sent. In x⟨y⟩ , this data is y . Similarly, if an input expects to receive data, one or more boundvariables will act as place-holders to be substituted by data, when it arrives. In x(v) , v plays that role. The choice ofthe kind of data that can be exchanged in an interaction is one of the key features that distinguishes different processcalculi.

61.2.3 Sequential composition

Sometimes interactions must be temporally ordered. For example, it might be desirable to specify algorithms such as:first receive some data on x and then send that data on y . Sequential composition can be used for such purposes. It iswell known from other models of computation. In process calculi, the sequentialisation operator is usually integratedwith input or output, or both. For example, the process x(v) · P will wait for an input on x . Only when this inputhas occurred will the process P be activated, with the received data through x substituted for identifier v .

61.2.4 Reduction semantics

The key operational reduction rule, containing the computational essence of process calculi, can be given solely interms of parallel composition, sequentialization, input, and output. The details of this reduction vary among thecalculi, but the essence remains roughly the same. The reduction rule is:

x⟨y⟩ · P | x(v) ·Q −→ P | Q[y/v]

The interpretation of this reduction rule is:

1. The process x⟨y⟩ ·P sends a message, here y , along the channel x . Dually, the process x(v) ·Q receives thatmessage on channel x .

2. Once the message has been sent, x⟨y⟩ · P becomes the process P , while x(v) ·Q becomes the process Q[y/v], which is Q with the place-holder v substituted by y , the data received on x .

The class of processes that P is allowed to range over as the continuation of the output operation substantially influ-ences the properties of the calculus.

https://en.wikipedia.org/wiki/%CE%A0-calculus




61.2.5 Hiding

Processes do not limit the number of connections that can be made at a given interaction point. But interaction pointsallow interference (i.e. interaction). For the synthesis of compact, minimal and compositional systems, the ability torestrict interference is crucial. Hiding operations allow control of the connections made between interaction pointswhen composing agents in parallel. Hiding can be denoted in a variety of ways. For example, in the π -calculus thehiding of a name x in P can be expressed as (ν x)P , while in CSP it might be written as P \ {x} .

61.2.6 Recursion and replication

The operations presented so far describe only finite interaction and are consequently insufficient for full computability,which includes non-terminating behaviour. Recursion and replication are operations that allow finite descriptionsof infinite behaviour. Recursion is well known from the sequential world. Replication !P can be understood asabbreviating the parallel composition of a countably infinite number of P processes:

!P = P |!P

61.2.7 Null process

Process calculi generally also include a null process (variously denoted as nil , 0 , STOP , δ , or some other appropriatesymbol) which has no interaction points. It is utterly inactive and its sole purpose is to act as the inductive anchor ontop of which more interesting processes can be generated.

61.3 Discrete and continuous process algebra

Process algebra has been studied for discrete time and continuous time (real time or dense time).[4]

61.4 History

In the first half of the 20th century, various formalisms were proposed to capture the informal concept of a com-putable function, with μ-recursive functions, Turing Machines and the lambda calculus possibly being the best-knownexamples today. The surprising fact that they are essentially equivalent, in the sense that they are all encodable intoeach other, supports the Church-Turing thesis. Another shared feature is more rarely commented on: they all aremost readily understood as models of sequential computation. The subsequent consolidation of computer science re-quired a more subtle formulation of the notion of computation, in particular explicit representations of concurrencyand communication. Models of concurrency such as the process calculi, Petri nets in 1962, and the Actor model in1973 emerged from this line of enquiry.Research on process calculi began in earnest with Robin Milner's seminal work on the Calculus of CommunicatingSystems (CCS) during the period from 1973 to 1980. C.A.R. Hoare's Communicating Sequential Processes (CSP)first appeared in 1978, and was subsequently developed into a full-fledged process calculus during the early 1980s.There was much cross-fertilization of ideas between CCS and CSP as they developed. In 1982 Jan Bergstra andJan Willem Klop began work on what came to be known as the Algebra of Communicating Processes (ACP), andintroduced the term process algebra to describe their work.[1] CCS, CSP, and ACP constitute the three major branchesof the process calculi family: the majority of the other process calculi can trace their roots to one of these three calculi.

61.5 Current research

Various process calculi have been studied and not all of them fit the paradigm sketched here. The most prominentexample may be the ambient calculus. This is to be expected as process calculi are an active field of study. Currentlyresearch on process calculi focuses on the following problems.

https://en.wikipedia.org/wiki/Communicating_sequential_processes

https://en.wikipedia.org/wiki/%CE%9C-recursive_function

https://en.wikipedia.org/wiki/Turing_Machine

https://en.wikipedia.org/wiki/Lambda_calculus







https://en.wikipedia.org/wiki/C.A.R._Hoare


https://en.wikipedia.org/wiki/Jan_Bergstra

https://en.wikipedia.org/wiki/Jan_Willem_Klop



61.6. SOFTWARE IMPLEMENTATIONS 229

• Developing new process calculi for better modeling of computational phenomena.

• Finding well-behaved subcalculi of a given process calculus. This is valuable because (1) most calculi are fairlywild in the sense that they are rather general and not much can be said about arbitrary processes; and (2)computational applications rarely exhaust the whole of a calculus. Rather they use only processes that are veryconstrained in form. Constraining the shape of processes is mostly studied by way of type systems.

• Logics for processes that allow one to reason about (essentially) arbitrary properties of processes, following theideas of Hoare logic.

• Behavioural theory: what does it mean for two processes to be the same? How can we decide whether twoprocesses are different or not? Can we find representatives for equivalence classes of processes? Generally,processes are considered to be the same if no context, that is other processes running in parallel, can detect adifference. Unfortunately, making this intuition precise is subtle and mostly yields unwieldy characterisations ofequality (which in most cases must also be undecidable, as a consequence of the halting problem). Bisimulationsare a technical tool that aids reasoning about process equivalences.

• Expressivity of calculi. Programming experience shows that certain problems are easier to solve in somelanguages than in others. This phenomenon calls for a more precise characterisation of the expressivity ofcalculi modeling computation than that afforded by the Church-Turing thesis. One way of doing this is toconsider encodings between two formalisms and see what properties encodings can potentially preserve. Themore properties can be preserved, the more expressive the target of the encoding is said to be. For processcalculi, the celebrated results are that the synchronous π -calculus is more expressive than its asynchronousvariant, has the same expressive power as the higher-order π -calculus, but is less than the ambient calculus.

• Using process calculus to model biological systems (stochasticπ -calculus, BioAmbients, Beta Binders, BioPEPA,Brane calculus). It is thought by some that the compositionality offered by process-theoretic tools can help bi-ologists to organise their knowledge more formally.

61.6 Software implementations

The ideas behind process algebra have given rise to several tools including:

• CADP

• Concurrency Workbench

• mCRL2 toolset

61.7 Relationship to other models of concurrency

The history monoid is the free object that is generically able to represent the histories of individual communicatingprocesses. A process calculus is then a formal language imposed on a history monoid in a consistent fashion.[5] Thatis, a history monoid can only record a sequence of events, with synchronization, but does not specify the allowedstate transitions. Thus, a process calculus is to a history monoid what a formal language is to a free monoid (a formallanguage is a subset of the set of all possible finite-length strings of an alphabet generated by the Kleene star).The use of channels for communication is one of the features distinguishing the process calculi from other models ofconcurrency, such as Petri nets and the Actor model (see Actor model and process calculi). One of the fundamentalmotivations for including channels in the process calculi was to enable certain algebraic techniques, thereby makingit easier to reason about processes algebraically.

61.8 See also• Stochastic probe


https://en.wikipedia.org/wiki/Hoare_logic

https://en.wikipedia.org/wiki/Halting_problem




https://en.wikipedia.org/wiki/Compositionality

https://en.wikipedia.org/wiki/CADP

http://homepages.inf.ed.ac.uk/perdita/cwb

http://www.mcrl2.org/

https://en.wikipedia.org/wiki/History_monoid


https://en.wikipedia.org/wiki/Formal_language




https://en.wikipedia.org/wiki/Concurrent_computing



https://en.wikipedia.org/wiki/Actor_model_and_process_calculi

https://en.wikipedia.org/wiki/Stochastic_probe


61.9 References[1] Baeten, J.C.M. (2004). “A brief history of process algebra” (PDF). Rapport CSR 04-02 (Vakgroep Informatica, Technische

Universiteit Eindhoven).

[2] Pierce, Benjamin. “Foundational Calculi for Programming Languages”. The Computer Science and Engineering Handbook.CRC Press. pp. 2190–2207. ISBN 0-8493-2909-4.

[3] Baeten, J.C.M.; Bravetti, M. (August 2005). “A Generic Process Algebra”. Algebraic Process Calculi: The First TwentyFive Years and Beyond (BRICS Notes Series NS-05-3). Bertinoro, Forl`ı, Italy: BRICS, Department of Computer Science,University of Aarhus. Retrieved 2007-12-29.

[4] Baeten, J. C. M.; Middelburg, C. A. “Process algebra with timing: Real time and discrete time”. CiteSeerX: 10 .1 .1 .42 .729.

[5] Mazurkiewicz, Antoni (1995). “Introduction to Trace Theory”. In Diekert, V.; Rozenberg, G. The Book of Traces(POSTSCRIPT). Singapore: World Scientific. pp. 3–41. ISBN 981-02-2058-8.

61.10 Further reading• Matthew Hennessy: Algebraic Theory of Processes, The MIT Press, ISBN 0-262-08171-7.

• C. A. R. Hoare: Communicating Sequential Processes, Prentice Hall, ISBN 0-13-153289-8.

• This book has been updated by Jim Davies at the Oxford University Computing Laboratory and the newedition is available for download as a PDF file at the Using CSP website.

• Robin Milner: A Calculus of Communicating Systems, Springer Verlag, ISBN 0-387-10235-3.

• Robin Milner: Communicating and Mobile Systems: the Pi-Calculus, Springer Verlag, ISBN 0-521-65869-1.

• Andrew Mironov: Theory of processes

http://alexandria.tue.nl/extra1/wskrap/publichtml/200402.pdf

https://en.wikipedia.org/wiki/Benjamin_C._Pierce



http://www.brics.dk/NS/05/3/



http://www.ipipan.waw.pl/~amaz/papers.htm/trbook.ps



https://en.wikipedia.org/wiki/Matthew_Hennessy

https://en.wikipedia.org/wiki/The_MIT_Press


https://en.wikipedia.org/wiki/C._A._R._Hoare

https://en.wikipedia.org/wiki/Prentice_Hall


https://en.wikipedia.org/wiki/Oxford_University_Computing_Laboratory

https://en.wikipedia.org/wiki/Portable_Document_Format

http://www.usingcsp.com/





https://en.wikipedia.org/wiki/Andrew_Mironov

http://arxiv.org/abs/1009.2259

Chapter 62

Series-parallel partial order

Series composition

Parallel composition

A series-parallel partial order, shown as a Hasse diagram.

In order-theoretic mathematics, a series-parallel partial order is a partially ordered set built up from smaller series-

231




232 CHAPTER 62. SERIES-PARALLEL PARTIAL ORDER

parallel partial orders by two simple composition operations.[1][2]

The series-parallel partial orders may be characterized as the N-free finite partial orders; they have order dimension atmost two.[1][3] They include weak orders and the reachability relationship in directed trees and directed series-parallelgraphs.[2][3] The comparability graphs of series-parallel partial orders are cographs.[2][4]

Series-parallel partial orders have been applied in job shop scheduling,[5] machine learning of event sequencing intime series data,[6] transmission sequencing of multimedia data,[7] and throughput maximization in dataflow pro-gramming.[8]

Series-parallel partial orders have also been called multitrees;[4] however, that name is ambiguous: multitrees alsorefer to partial orders with no four-element diamond suborder[9] and to other structures formed from multiple trees.

62.1 Definition

Consider P and Q, two partially ordered sets. The series composition of P and Q, written P; Q,[7] P * Q,[2] or P ⧀Q,[1]is the partially ordered set whose elements are the disjoint union of the elements of P and Q. In P; Q, two elementsx and y that both belong to P or that both belong to Q have the same order relation that they do in P or Q respectively.However, for every pair x, y where x belongs to P and y belongs to Q, there is an additional order relation x ≤ y in theseries composition. Series composition is an associative operation: one can write P; Q; R as the series compositionof three orders, without ambiguity about how to combine them pairwise, because both of the parenthesizations (P;Q); R and P; (Q; R) describe the same partial order. However, it is not a commutative operation, because switchingthe roles of P and Q will produce a different partial order that reverses the order relations of pairs with one elementin P and one in Q.[1]

The parallel composition of P and Q, written P || Q,[7] P + Q,[2] or P ⊕ Q,[1] is defined similarly, from the disjointunion of the elements in P and the elements in Q, with pairs of elements that both belong to P or both to Q havingthe same order as they do in P or Q respectively. In P || Q, a pair x, y is incomparable whenever x belongs to P and ybelongs to Q. Parallel composition is both commutative and associative.[1]

The class of series-parallel partial orders is the set of partial orders that can be built up from single-element partialorders using these two operations. Equivalently, it is the smallest set of partial orders that includes the single-elementpartial order and is closed under the series and parallel composition operations.[1][2]

A weak order is the series parallel partial order obtained from a sequence of composition operations in which allof the parallel compositions are performed first, and then the results of these compositions are combined using onlyseries compositions.[2]

62.2 Forbidden suborder characterization

The partial order N with the four elements a, b, c, and d and exactly the three order relations a ≤ b ≥ c ≤ d is anexample of a fence or zigzag poset; its Hasse diagram has the shape of the capital letter “N”. It is not series-parallel,because there is no way of splitting it into the series or parallel composition of two smaller partial orders. A partialorder P is said to be N-free if there does not exist a set of four elements in P such that the restriction of P to thoseelements is order-isomorphic to N. The series-parallel partial orders are exactly the nonempty finite N-free partialorders.[1][2][3]

It follows immediately from this (although it can also be proven directly) that any nonempty restriction of a series-parallel partial order is itself a series-parallel partial order.[1]

62.3 Order dimension

The order dimension of a partial order P is the minimum size of a realizer of P, a set of linear extensions of P withthe property that, for every two distinct elements x and y of P, x ≤ y in P if and only if x has an earlier position thany in every linear extension of the realizer. Series-parallel partial orders have order dimension at most two. If P andQ have realizers {L1, L2} and {L3, L4}, respectively, then {L1L3, L2L4} is a realizer of the series composition P; Q,and {L1L3, L4L2} is a realizer of the parallel composition P || Q.[2][3] A partial order is series-parallel if and only ifit has a realizer in which one of the two permutations is the identity and the other is a separable permutation.


https://en.wikipedia.org/wiki/Weak_order


https://en.wikipedia.org/wiki/Tree_(graph_theory)

https://en.wikipedia.org/wiki/Series-parallel_graph

https://en.wikipedia.org/wiki/Series-parallel_graph


https://en.wikipedia.org/wiki/Cograph

https://en.wikipedia.org/wiki/Job_shop_scheduling

https://en.wikipedia.org/wiki/Machine_learning

https://en.wikipedia.org/wiki/Time_series

https://en.wikipedia.org/wiki/Multimedia

https://en.wikipedia.org/wiki/Dataflow_programming

https://en.wikipedia.org/wiki/Dataflow_programming

https://en.wikipedia.org/wiki/Multitree

https://en.wikipedia.org/wiki/Disjoint_union

https://en.wikipedia.org/wiki/Associative_operation

https://en.wikipedia.org/wiki/Commutative_operation

https://en.wikipedia.org/wiki/Closure_(mathematics)

https://en.wikipedia.org/wiki/Weak_order

https://en.wikipedia.org/wiki/Fence_(mathematics)




https://en.wikipedia.org/wiki/Separable_permutation

62.4. CONNECTIONS TO GRAPH THEORY 233

It is known that a partial order P has order dimension two if and only if there exists a conjugate order Q on thesame elements, with the property that any two distinct elements x and y are comparable on exactly one of these twoorders. In the case of series parallel partial orders, a conjugate order that is itself series parallel may be obtained byperforming a sequence of composition operations in the same order as the ones defining P on the same elements,but performing a series composition for each parallel composition in the decomposition of P and vice versa. Morestrongly, although a partial order may have many different conjugates, every conjugate of a series parallel partial ordermust itself be series parallel.[2]

62.4 Connections to graph theory

Any partial order may be represented (usually in more than one way) by a directed acyclic graph in which there is apath from x to y whenever x and y are elements of the partial order with x ≤ y. The graphs that represent series-parallelpartial orders in this way have been called vertex series parallel graphs, and their transitive reductions (the graphsof the covering relations of the partial order) are called minimal vertex series parallel graphs.[3] Directed trees and(two-terminal) series parallel graphs are examples of minimal vertex series parallel graphs; therefore, series parallelpartial orders may be used to represent reachability relations in directed trees and series parallel graphs.[2][3]

The comparability graph of a partial order is the undirected graph with a vertex for each element and an undirectededge for each pair of distinct elements x, y with either x ≤ y or y ≤ x. That is, it is formed from a minimal vertexseries parallel graph by forgetting the orientation of each edge. The comparability graph of a series-partial order is acograph: the series and parallel composition operations of the partial order give rise to operations on the comparabilitygraph that form the disjoint union of two subgraphs or that connect two subgraphs by all possible edges; these twooperations are the basic operations from which cographs are defined. Conversely, every cograph is the comparabilitygraph of a series-parallel partial order. If a partial order has a cograph as its comparability graph, then it must be aseries-parallel partial order, because every other kind of partial order has an N suborder that would correspond to aninduced four-vertex path in its comparability graph, and such paths are forbidden in cographs.[2][4]

62.5 Computational complexity

It is possible to use the forbidden suborder characterization of series-parallel partial orders as a basis for an algorithmthat tests whether a given binary relation is a series-parallel partial order, in an amount of time that is linear inthe number of related pairs.[2][3] Alternatively, if a partial order is described as the reachability order of a directedacyclic graph, it is possible to test whether it is a series-parallel partial order, and if so compute its transitive closure,in time proportional to the number of vertices and edges in the transitive closure; it remains open whether the timeto recognize series-parallel reachability orders can be improved to be linear in the size of the input graph.[10]

If a series-parallel partial order is represented as an expression tree describing the series and parallel compositionoperations that formed it, then the elements of the partial order may be represented by the leaves of the expressiontree. A comparison between any two elements may be performed algorithmically by searching for the lowest commonancestor of the corresponding two leaves; if that ancestor is a parallel composition, the two elements are incomparable,and otherwise the order of the series composition operands determines the order of the elements. In this way, a series-parallel partial order on n elements may be represented in O(n) space with O(1) time to determine any comparisonvalue.[2]

It is NP-complete to test, for two given series-parallel partial orders P and Q, whether P contains a restriction iso-morphic to Q.[3]

Although the problem of counting the number of linear extensions of an arbitrary partial order is #P-complete,[11] itmay be solved in polynomial time for series-parallel partial orders. Specifically, if L(P) denotes the number of linearextensions of a partial order P, then L(P; Q) = L(P)L(Q) and

L(P ||Q) =(|P |+ |Q|)!|P |!|Q|!

L(P )L(Q),

so the number of linear extensions may be calculated using an expression tree with the same form as the decompositiontree of the given series-parallel order.[2]




https://en.wikipedia.org/wiki/Series_parallel_graph



https://en.wikipedia.org/wiki/Orientation_(graph_theory)





https://en.wikipedia.org/wiki/Expression_tree

https://en.wikipedia.org/wiki/Lowest_common_ancestor

https://en.wikipedia.org/wiki/Lowest_common_ancestor

https://en.wikipedia.org/wiki/NP-complete

https://en.wikipedia.org/wiki/Sharp-P-complete

234 CHAPTER 62. SERIES-PARALLEL PARTIAL ORDER

62.6 Applications

Mannila & Meek (2000) use series-parallel partial orders as a model for the sequences of events in time series data.They describe machine learning algorithms for inferring models of this type, and demonstrate its effectiveness atinferring course prerequisites from student enrollment data and at modeling web browser usage patterns.[6]

Amer et al. (1994) argue that series-parallel partial orders are a good fit for modeling the transmission sequencingrequirements of multimedia presentations. They use the formula for computing the number of linear extensions of aseries-parallel partial order as the basis for analyzing multimedia transmission algorithms.[7]

Choudhary et al. (1994) use series-parallel partial orders to model the task dependencies in a dataflow model ofmassive data processing for computer vision. They show that, by using series-parallel orders for this problem, it ispossible to efficiently construct an optimized schedule that assigns different tasks to different processors of a parallelcomputing system in order to optimize the throughput of the system.[8]

A class of orderings somewhat more general than series-parallel partial orders is provided by PQ trees, data structuresthat have been applied in algorithms for testing whether a graph is planar and recognizing interval graphs.[12] A Pnode of a PQ tree allows all possible orderings of its children, like a parallel composition of partial orders, while a Qnode requires the children to occur in a fixed linear ordering, like a series composition of partial orders. However,unlike series-parallel partial orders, PQ trees allow the linear ordering of any Q node to be reversed.

62.7 See also• Series and parallel circuits

62.8 References[1] Bechet, Denis; De Groote, Philippe; Retoré, Christian (1997), “A complete axiomatisation for the inclusion of series-

parallel partial orders”, Rewriting Techniques and Applications, Lecture Notes in Computer Science 1232, Springer-Verlag,pp. 230–240, doi:10.1007/3-540-62950-5_74.

[2] Möhring, Rolf H. (1989), “Computationally tractable classes of ordered sets”, in Rival, Ivan, Algorithms and Order: Pro-ceedings of the NATO Advanced Study Institute on Algorithms and Order, Ottawa, Canada, May 31-June 13, 1987, NATOScience Series C 255, Springer-Verlag, pp. 105–194, ISBN 978-0-7923-0007-6.

[3] Valdes, Jacobo; Tarjan, Robert E.; Lawler, Eugene L. (1982), “The recognition of series parallel digraphs”, SIAM Journalon Computing 11 (2): 298–313, doi:10.1137/0211023.

[4] Jung, H. A. (1978), “On a class of posets and the corresponding comparability graphs”, Journal of Combinatorial Theory,Series B 24 (2): 125–133, doi:10.1016/0095-8956(78)90013-8, MR 0491356.

[5] Lawler, Eugene L. (1978), “Sequencing jobs to minimize total weighted completion time subject to precedence constraints”,Annals of Discrete Mathematics 2: 75–90, doi:10.1016/S0167-5060(08)70323-6, MR 0495156.

[6] Mannila, Heikki; Meek, Christopher (2000), “Global partial orders from sequential data”, Proc. 6th ACM SIGKDD Inter-national Conference on Knowledge Discovery and Data Mining (KDD 2000), pp. 161–168, doi:10.1145/347090.347122.

[7] Amer, Paul D.; Chassot, Christophe; Connolly, Thomas J.; Diaz, Michel; Conrad, Phillip (1994), “Partial-order transportservice for multimedia and other applications”, IEEE/ACMTransactions onNetworking 2 (5): 440–456, doi:10.1109/90.336326.

[8] Choudhary, A. N.; Narahari, B.; Nicol, D. M.; Simha, R. (1994), “Optimal processor assignment for a class of pipelinedcomputations”, IEEE Transactions on Parallel and Distributed Systems 5 (4): 439–445, doi:10.1109/71.273050.

[9] Furnas, George W.; Zacks, Jeff (1994), “Multitrees: enriching and reusing hierarchical structure”, Proc. SIGCHI conferenceon Human Factors in Computing Systems (CHI '94), pp. 330–336, doi:10.1145/191666.191778.

[10] Ma, Tze-Heng; Spinrad, Jeremy (1991), “Transitive closure for restricted classes of partial orders”, Order 8 (2): 175–183,doi:10.1007/BF00383402.

[11] Brightwell, Graham R.; Winkler, Peter (1991), “Counting linear extensions”, Order 8 (3): 225–242, doi:10.1007/BF00383444.

[12] Booth, Kellogg S.; Lueker, George S. (1976), “Testing for the consecutive ones property, interval graphs, and graphplanarity using PQ-tree algorithms”, Journal of Computer and System Sciences 13 (3): 335–379, doi:10.1016/S0022-0000(76)80045-1.

https://en.wikipedia.org/wiki/Series-parallel_partial_order#CITEREFMannilaMeek2000

https://en.wikipedia.org/wiki/Time_series

https://en.wikipedia.org/wiki/Machine_learning

https://en.wikipedia.org/wiki/Series-parallel_partial_order#CITEREFAmerChassotConnollyDiaz1994

https://en.wikipedia.org/wiki/Multimedia

https://en.wikipedia.org/wiki/Series-parallel_partial_order#CITEREFChoudharyNarahariNicolSimha1994

https://en.wikipedia.org/wiki/Dataflow

https://en.wikipedia.org/wiki/Computer_vision

https://en.wikipedia.org/wiki/Parallel_computing

https://en.wikipedia.org/wiki/Parallel_computing

https://en.wikipedia.org/wiki/PQ_tree

https://en.wikipedia.org/wiki/Planar_graph


https://en.wikipedia.org/wiki/Series_and_parallel_circuits


https://dx.doi.org/10.1007%252F3-540-62950-5_74

https://en.wikipedia.org/wiki/Ivan_Rival



https://en.wikipedia.org/wiki/Robert_Tarjan

https://en.wikipedia.org/wiki/Eugene_Lawler

https://en.wikipedia.org/wiki/SIAM_Journal_on_Computing

https://en.wikipedia.org/wiki/SIAM_Journal_on_Computing


https://dx.doi.org/10.1137%252F0211023

https://en.wikipedia.org/wiki/Journal_of_Combinatorial_Theory


https://dx.doi.org/10.1016%252F0095-8956%252878%252990013-8



https://en.wikipedia.org/wiki/Eugene_Lawler


https://dx.doi.org/10.1016%252FS0167-5060%252808%252970323-6



https://en.wikipedia.org/wiki/Heikki_Mannila


https://dx.doi.org/10.1145%252F347090.347122


https://dx.doi.org/10.1109%252F90.336326


https://dx.doi.org/10.1109%252F71.273050


https://dx.doi.org/10.1145%252F191666.191778



https://en.wikipedia.org/wiki/Peter_Winkler




https://en.wikipedia.org/wiki/Journal_of_Computer_and_System_Sciences




Chapter 63

Surjective function

“Onto” redirects here. For other uses, see wikt:onto.In mathematics, a function f from a set X to a set Y is surjective (or onto), or a surjection, if every element y in Y

X1

2

3

4

YD

B

C

A surjective function from domain X to codomain Y. The function is surjective because every point in the codomain is the value off(x) for at least one point x in the domain.

235

https://en.wiktionary.org/wiki/onto






236 CHAPTER 63. SURJECTIVE FUNCTION

has a corresponding element x in X such that f(x) = y. The function f may map more than one element of X to thesame element of Y.The term surjective and the related terms injective and bijective were introduced by Nicolas Bourbaki,[1] the pseudonymfor a group of mainly French 20th-century mathematicians who wrote a series of books presenting an exposition ofmodern advanced mathematics, beginning in 1935. The French prefix sur means over or above and relates to the factthat the image of the domain of a surjective function completely covers the function’s codomain.

63.1 Definition

For more details on notation, see Function (mathematics) § Notation.

A surjective function is a function whose image is equal to its codomain. Equivalently, a function f with domainX and codomain Y is surjective if for every y in Y there exists at least one x in X with f(x) = y . Surjections aresometimes denoted by a two-headed rightwards arrow (U+21A0 ↠ rightwards two headed arrow),[2] as in f : X ↠Y.Symbolically,

If f : X → Y , then f is said to be surjective if

∀y ∈ Y, ∃x ∈ X, f(x) = y

63.2 Examples

For any set X, the identity function idX on X is surjective.The function f : Z → {0,1} defined by f(n) = n mod 2 (that is, even integers are mapped to 0 and odd integers to 1)is surjective.The function f : R → R defined by f(x) = 2x + 1 is surjective (and even bijective), because for every real number ywe have an x such that f(x) = y: an appropriate x is (y − 1)/2.The function f : R → R defined by f(x) = x3 − 3x is surjective, because the pre-image of any real number y is thesolution set of the cubic polynomial equation x3 − 3x − y = 0 and every cubic polynomial with real coefficients has atleast one real root. However, this function is not injective (and hence not bijective) since e.g. the pre-image of y = 2is {x = −1, x = 2}. (In fact, the pre-image of this function for every y, −2 ≤ y ≤ 2 has more than one element.)The function g : R → R defined by g(x) = x2 is not surjective, because there is no real number x such that x2 = −1.However, the function g : R → R0

+ defined by g(x) = x2 (with restricted codomain) is surjective because for every yin the nonnegative real codomain Y there is at least one x in the real domain X such that x2 = y.The natural logarithm function ln : (0,+∞) → R is a surjective and even bijective mapping from the set of positivereal numbers to the set of all real numbers. Its inverse, the exponential function, is not surjective as its range is the setof positive real numbers and its domain is usually defined to be the set of all real numbers. The matrix exponentialis not surjective when seen as a map from the space of all n×n matrices to itself. It is, however, usually defined as amap from the space of all n×n matrices to the general linear group of degree n, i.e. the group of all n×n invertiblematrices. Under this definition the matrix exponential is surjective for complex matrices, although still not surjectivefor real matrices.The projection from a cartesian product A × B to one of its factors is surjective unless the other factor is empty.In a 3D video game vectors are projected onto a 2D flat screen by means of a surjective function.

63.3 Properties

A function is bijective if and only if it is both surjective and injective.




https://en.wikipedia.org/wiki/France

https://en.wikipedia.org/wiki/Mathematician

https://en.wikipedia.org/wiki/Prefix

https://en.wiktionary.org/wiki/sur#French





https://en.wikipedia.org/wiki/Function(mathematics)




https://en.wikipedia.org/wiki/Unicode





https://en.wikipedia.org/wiki/Odd_number








https://en.wikipedia.org/wiki/Matrix_exponential


https://en.wikipedia.org/wiki/General_linear_group


https://en.wikipedia.org/wiki/Invertible_matrix

https://en.wikipedia.org/wiki/Invertible_matrix





63.3. PROPERTIES 237

X Y

f(x)

f : X → Y

x

A non-surjective function from domain X to codomain Y. The smaller oval inside Y is the image (also called range) of f. Thisfunction is not surjective, because the image does not fill the whole codomain. In other words, Y is colored in a two-step process:First, for every x in X, the point f(x) is colored yellow; Second, all the rest of the points in Y, that are not yellow, are colored blue.The function f is surjective only if there are no blue points.

If (as is often done) a function is identified with its graph, then surjectivity is not a property of the function itself, butrather a relationship between the function and its codomain. Unlike injectivity, surjectivity cannot be read off of thegraph of the function alone.

63.3.1 Surjections as right invertible functions

The function g : Y → X is said to be a right inverse of the function f : X → Y if f(g(y)) = y for every y in Y (g can beundone by f). In other words, g is a right inverse of f if the composition f o g of g and f in that order is the identityfunction on the domain Y of g. The function g need not be a complete inverse of f because the composition in theother order, g o f, may not be the identity function on the domain X of f. In other words, f can undo or "reverse" g,but cannot necessarily be reversed by it.Every function with a right inverse is necessarily a surjection. The proposition that every surjective function has aright inverse is equivalent to the axiom of choice.If f : X → Y is surjective and B is a subset of Y, then f(f −1(B)) = B. Thus, B can be recovered from its preimagef −1(B).For example, in the first illustration, there is some function g such that g(C) = 4. There is also some function f suchthat f(4) = C. It doesn't matter that g(C) can also equal 3; it only matters that f “reverses” g.

• Another surjective function. (This one happens to be a bijection)

• A non-surjective function. (This one happens to be an injection)

• Surjective composition: the first function need not be surjective.






https://en.wikipedia.org/wiki/Inverse_function#Left_and_right_inverses











x

y

x

y

1 X 1 Y : f 2 X 2 Y : f

2 X x 1 X x

x f y

f im Y y

Y y

f im

Y X : f x f y

X x

Interpretation for surjective functions in the Cartesian plane, defined by the mapping f : X → Y, where y = f(x), X = domain offunction, Y = range of function. Every element in the range is mapped onto from an element in the domain, by the rule f. Theremay be a number of domain elements which map to the same range element. That is, every y in Y is mapped from an element x inX, more than one x can map to the same y. Left: Only one domain is shown which makes f surjective. Right: two possible domainsX1 and X2 are shown.

x

y

X x 0

Y X : f x f y

X x

Y y

f im

Y y 0

x

y

X x 1

Y y 2

Y y 1

X x 2

X x 3

Y y 3

X x

1 X 1 Y : f 2 X 2 Y : f

Y y

f im

2 X x 1 X x

x f y

Non-surjective functions in the Cartesian plane. Although some parts of the function are surjective, where elements y in Y do havea value x in X such that y = f(x), some parts are not. Left: There is y0 in Y, but there is no x0 in X such that y0 = f(x0). Right:There are y1, y2 and y3 in Y, but there are no x1, x2, and x3 in X such that y1 = f(x1), y2 = f(x2), and y3 = f(x3).

63.3.2 Surjections as epimorphisms

A function f : X → Y is surjective if and only if it is right-cancellative:[3] given any functions g,h : Y → Z, whenever go f = h o f, then g = h. This property is formulated in terms of functions and their composition and can be generalizedto the more general notion of the morphisms of a category and their composition. Right-cancellative morphisms arecalled epimorphisms. Specifically, surjective functions are precisely the epimorphisms in the category of sets. The

https://en.wikipedia.org/wiki/Right-cancellative






63.4. SEE ALSO 239

prefix epi is derived from the Greek preposition ἐπί meaning over, above, on.Any morphism with a right inverse is an epimorphism, but the converse is not true in general. A right inverse g of amorphism f is called a section of f. A morphism with a right inverse is called a split epimorphism.

63.3.3 Surjections as binary relations

Any function with domain X and codomain Y can be seen as a left-total and right-unique binary relation between Xand Y by identifying it with its function graph. A surjective function with domain X and codomain Y is then a binaryrelation between X and Y that is right-unique and both left-total and right-total.

63.3.4 Cardinality of the domain of a surjection

The cardinality of the domain of a surjective function is greater than or equal to the cardinality of its codomain: If f: X → Y is a surjective function, then X has at least as many elements as Y, in the sense of cardinal numbers. (Theproof appeals to the axiom of choice to show that a function g : Y → X satisfying f(g(y)) = y for all y in Y exists. gis easily seen to be injective, thus the formal definition of |Y | ≤ |X| is satisfied.)Specifically, if both X and Y are finite with the same number of elements, then f : X → Y is surjective if and only iff is injective.

63.3.5 Composition and decomposition

The composite of surjective functions is always surjective: If f and g are both surjective, and the codomain of gis equal to the domain of f, then f o g is surjective. Conversely, if f o g is surjective, then f is surjective (but g,the function applied first, need not be). These properties generalize from surjections in the category of sets to anyepimorphisms in any category.Any function can be decomposed into a surjection and an injection: For any function h : X → Z there exist a surjectionf : X → Y and an injection g : Y → Z such that h = g o f. To see this, define Y to be the sets h −1(z) where z is inZ. These sets are disjoint and partition X. Then f carries each x to the element of Y which contains it, and g carrieseach element of Y to the point in Z to which h sends its points. Then f is surjective since it is a projection map, andg is injective by definition.

63.3.6 Induced surjection and induced bijection

Any function induces a surjection by restricting its codomain to its range. Any surjective function induces a bijectiondefined on a quotient of its domain by collapsing all arguments mapping to a given fixed image. More precisely, everysurjection f : A → B can be factored as a projection followed by a bijection as follows. Let A/~ be the equivalenceclasses of A under the following equivalence relation: x ~ y if and only if f(x) = f(y). Equivalently, A/~ is the set ofall preimages under f. Let P(~) : A → A/~ be the projection map which sends each x in A to its equivalence class[x]~, and let fP : A/~ → B be the well-defined function given by fP([x]~) = f(x). Then f = fP o P(~).

63.4 See also• Bijection, injection and surjection

• Cover (algebra)

• Covering map

• Enumeration

• Fiber bundle

• Index set

• Section (category theory)


https://en.wikipedia.org/wiki/Split_epimorphism

https://en.wikipedia.org/wiki/Left-total_relation

https://en.wikipedia.org/wiki/Right-unique_relation

https://en.wikipedia.org/wiki/Function_graph

https://en.wikipedia.org/wiki/Right-total_relation




https://en.wikipedia.org/wiki/Cardinal_number#Formal_definition














https://en.wikipedia.org/wiki/Projection_map

https://en.wikipedia.org/wiki/Bijection,_injection_and_surjection

https://en.wikipedia.org/wiki/Cover_(algebra)

https://en.wikipedia.org/wiki/Covering_map

https://en.wikipedia.org/wiki/Enumeration

https://en.wikipedia.org/wiki/Fiber_bundle




63.5 Notes[1] “Injection, Surjection and Bijection”, Earliest Uses of Some of the Words of Mathematics, Tripod.

[2] “Arrows – Unicode” (PDF). Retrieved 2013-05-11.

[3] Goldblatt, Robert (2006) [1984]. Topoi, the Categorial Analysis of Logic (Revised ed.). Dover Publications. ISBN 978-0-486-45026-1. Retrieved 2009-11-25.

63.6 References• Bourbaki, Nicolas (2004) [1968]. Theory of Sets. Springer. ISBN 978-3-540-22525-6.


http://www.unicode.org/charts/PDF/U2190.pdf

http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=Gold010&id=3

https://en.wikipedia.org/wiki/Dover_Publications







Chapter 64

Symmetric closure

In mathematics, the symmetric closure of a binary relation R on a set X is the smallest symmetric relation on X thatcontains R.For example, if X is a set of airports and xRy means “there is a direct flight from airport x to airport y", then thesymmetric closure of R is the relation “there is a direct flight either from x to y or from y to x". Or, if X is the set ofhumans (alive or dead) and R is the relation 'parent of', then the symmetric closure of R is the relation "x is a parentor a child of y".

64.1 Definition

The symmetric closure S of a relation R on a set X is given by

S = R ∪ {(x, y) : (y, x) ∈ R} .

In other words, the symmetric closure of R is the union of R with its inverse relation, R−1.

64.2 See also• Transitive closure

• Reflexive closure

64.3 References• Franz Baader and Tobias Nipkow, Term Rewriting and All That, Cambridge University Press, 1998, p. 8

241








https://en.wikipedia.org/wiki/Franz_Baader

https://en.wikipedia.org/wiki/Tobias_Nipkow

Chapter 65

Symmetric relation

In mathematics and other areas, a binary relation R over a set X is symmetric if it holds for all a and b in X that if ais related to b then b is related to a.In mathematical notation, this is:

∀a, b ∈ X, aRb⇒ bRa.

65.1 Examples

65.1.1 In mathematics

• “is equal to” (equality) (whereas “is less than” is not symmetric)

• “is comparable to”, for elements of a partially ordered set

• "... and ... are odd":

242








65.2. RELATIONSHIP TO ASYMMETRIC AND ANTISYMMETRIC RELATIONS 243

65.1.2 Outside mathematics

• “is married to” (in most legal systems)

• “is a fully biological sibling of”

• “is a homophone of”

65.2 Relationship to asymmetric and antisymmetric relations

By definition, a relation cannot be both symmetric and asymmetric (where if a is related to b, then b cannot be relatedto a (in the same way)). However, a relation can be neither symmetric nor asymmetric, which is the case for “is lessthan or equal to” and “preys on”).Symmetric and antisymmetric (where the only way a can be related to b and b be related to a is if a = b) are actuallyindependent of each other, as these examples show.

65.3 Additional aspects

A symmetric relation that is also transitive and reflexive is an equivalence relation.One way to conceptualize a symmetric relation in graph theory is that a symmetric relation is an edge, with the edge’stwo vertices being the two entities so related. Thus, symmetric relations and undirected graphs are combinatoriallyequivalent objects.

https://en.wikipedia.org/wiki/Homophone






244 CHAPTER 65. SYMMETRIC RELATION

65.4 See also• Symmetry in mathematics

• Symmetry

• Asymmetric relation

• Antisymmetric relation


https://en.wikipedia.org/wiki/Symmetry



Chapter 66

Ternary equivalence relation

In mathematics, a ternary equivalence relation is a kind of ternary relation analogous to a binary equivalencerelation. A ternary equivalence relation is symmetric, reflexive, and transitive. The classic example is the relationof collinearity among three points in Euclidean space. In an abstract set, a ternary equivalence relation determines acollection of equivalence classes or pencils that form a linear space in the sense of incidence geometry. In the sameway, a binary equivalence relation on a set determines a partition.

66.1 Definition

A ternary equivalence relation on a set X is a relation E ⊂ X3, written [a, b, c], that satisfies the following axioms:

1. Symmetry: If [a, b, c] then [b, c, a] and [c, b, a]. (Therefore also [a, c, b], [b, a, c], and [c, a, b].)

2. Reflexivity: [a, b, b]. Equivalently, if a, b, and c are not all distinct, then [a, b, c].

3. Transitivity: If a ≠ b and [a, b, c] and [a, b, d] then [b, c, d]. (Therefore also [a, c, d].)

66.2 References• Araújoa, João; Koniecznyc, Janusz (2007), “A method of finding automorphism groups of endomorphism

monoids of relational systems”, Discrete Mathematics 307: 1609–1620, doi:10.1016/j.disc.2006.09.029

• Bachmann, Friedrich, Aufbau der Geometrie aus dem Spiegelungsbegriff

• Karzel, Helmut (2007), “Loops related to geometric structures”, Quasigroups and Related Systems 15: 47−76

• Karzel, Helmut; Pianta, Silvia (2008), “Binary operations derived from symmetric permutation sets and appli-cations to absolute geometry”, Discrete Mathematics 308: 415–421, doi:10.1016/j.disc.2006.11.058

• Karzel, Helmut; Marchi, Mario; Pianta, Silvia (December 2010), “The defect in an invariant reflection struc-ture”, Journal of Geometry 99 (1-2): 67–87, doi:10.1007/s00022-010-0058-7

• Lingenberg, Rolf (1979), Metric planes and metric vector spaces, Wiley

• Rainich, G.Y. (1952), “Ternary relations in geometry and algebra”, Michigan Mathematical Journal 1 (2):97–111, doi:10.1307/mmj/1028988890

• Szmielew, Wanda (1981), On n-ary equivalence relations and their application to geometry, Warsaw: InstytutMatematyczny Polskiej Akademi Nauk

245






https://en.wikipedia.org/wiki/Collinearity


https://en.wikipedia.org/wiki/Pencil_(mathematics)

https://en.wikipedia.org/wiki/Linear_space_(geometry)

https://en.wikipedia.org/wiki/Incidence_geometry







https://dx.doi.org/10.1007%252Fs00022-010-0058-7


https://dx.doi.org/10.1307%252Fmmj%252F1028988890

http://eudml.org/doc/268578

Chapter 67

Ternary relation

In mathematics, a ternary relation or triadic relation is a finitary relation in which the number of places in therelation is three. Ternary relations may also be referred to as 3-adic, 3-ary, 3-dimensional, or 3-place.Just as a binary relation is formally defined as a set of pairs, i.e. a subset of the Cartesian product A × B of some setsA and B, so a ternary relation is a set of triples, forming a subset of the Cartesian product A × B × C of three sets A,B and C.An example of a ternary relation in elementary geometry is the collinearity of points.

67.1 Examples

67.1.1 Binary functions

Further information: Graph of a function and binary function

A function ƒ: A × B → C in two variables, taking values in two sets A and B, respectively, is formally a function thatassociates to every pair (a,b) in A × B an element ƒ(a, b) in C. Therefore its graph consists of pairs of the form ((a,b), ƒ(a, b)). Such pairs in which the first element is itself a pair are often identified with triples. This makes the graphof ƒ a ternary relation between A, B and C, consisting of all triples (a, b, ƒ(a, b)), for all a in A and b in B.

67.1.2 Cyclic orders

Main article: Cyclic order

Given any set A whose elements are arranged on a circle, one can define a ternary relation R on A, i.e. a subset of A3

= A × A × A, by stipulating that R(a, b, c) holds if and only if the elements a, b and c are pairwise different and whengoing from a to c in a clockwise direction one passes through b. For example if A = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,12 } represents the hours on a clock face, then R(8, 12, 4) holds and R(12, 8, 4) does not hold.

67.1.3 Betweenness relations

Main article: Betweenness relation

67.1.4 Congruence relation

Main article: Congruence modulo m

246





https://en.wikipedia.org/wiki/Line_(geometry)#Collinear_points


https://en.wikipedia.org/wiki/Binary_function

https://en.wikipedia.org/wiki/Cyclic_order

https://en.wikipedia.org/wiki/Clock_face

https://en.wikipedia.org/wiki/Betweenness_relation

https://en.wikipedia.org/wiki/Congruence_modulo_m


The ordinary congruence of arithmetics

a ≡ b (mod m)

which holds for three integers a, b, and m if and only if m divides a − b, formally may be considered as a ternaryrelation. However, usually, this instead is considered as a family of binary relations between the a and the b, indexedby the modulus m. For each fixed m, indeed this binary relation has some natural properties, like being an equivalencerelation; while the combined ternary relation in general is not studied as one relation.

67.1.5 Typing relation

Main article: Simply typed lambda calculus § Typing rules

A typing relation Γ ⊢ e : σ indicates that e is a term of type σ in context Γ , and is thus a ternary relation betweencontexts, terms and types.

67.2 Further reading• Myers, Dale (1997), “An interpretive isomorphism between binary and ternary relations”, in Mycielski, Jan;

Rozenberg, Grzegorz; Salomaa, Arto, Structures in Logic and Computer Science, Lecture Notes in ComputerScience 1261, Springer, pp. 84–105, doi:10.1007/3-540-63246-8_6, ISBN 3-540-63246-8

• Novák, Vítězslav (1996), “Ternary structures and partial semigroups”, Czechoslovak Mathematical Journal 46(1): 111–120, hdl:10338.dmlcz/127275

• Novák, Vítězslav; Novotný, Miroslav (1989), “Transitive ternary relations and quasiorderings”, ArchivumMathematicum 25 (1–2): 5–12, hdl:10338.dmlcz/107333

• Novák, Vítězslav; Novotný, Miroslav (1992), “Binary and ternary relations”, Mathematica Bohemica 117 (3):283–292, hdl:10338.dmlcz/126278

• Novotný, Miroslav (1991), “Ternary structures and groupoids”, Czechoslovak Mathematical Journal 41 (1):90–98, hdl:10338.dmlcz/102437

• Šlapal, Josef (1993), “Relations and topologies”, CzechoslovakMathematical Journal 43 (1): 141–150, hdl:10338.dmlcz/128381





https://en.wikipedia.org/wiki/Simply_typed_lambda_calculus#Typing_rules


https://dx.doi.org/10.1007%252F3-540-63246-8_6



https://en.wikipedia.org/wiki/Handle_System

http://hdl.handle.net/10338.dmlcz%252F127275









Chapter 68

Tolerance relation

In mathematics, a tolerance relation is a relation that is reflexive and symmetric. It does not need to be transitive.

68.1 External links• Gerasin, S. N., Shlyakhov, V. V., and Yakovlev, S. V. 2008. Set coverings and tolerance relations. Cybernetics

and Sys. Anal. 44, 3 (May 2008), 333–340. doi:10.1007/s10559-008-9007-y

• Hryniewiecki, K. 1991, Relations of Tolerance

FORMALIZED MATHEMATICS, Vol. 2, No. 1, January–February 1991.

248






https://dx.doi.org/10.1007%252Fs10559-008-9007-y

Chapter 69

Total order

In mathematics, a linear order, total order, simple order, or (non-strict) ordering is a binary relation (here denotedby infix ≤) on some set X which is transitive, antisymmetric, and total. A set paired with a total order is called a totallyordered set, a linearly ordered set, a simply ordered set, or a chain.If X is totally ordered under ≤, then the following statements hold for all a, b and c in X:

If a ≤ b and b ≤ a then a = b (antisymmetry);If a ≤ b and b ≤ c then a ≤ c (transitivity);a ≤ b or b ≤ a (totality).

Antisymmetry eliminates uncertain cases when both a precedes b and b precedes a.[1] A relation having the property of“totality” means that any pair of elements in the set of the relation are comparable under the relation. This also meansthat the set can be diagrammed as a line of elements, giving it the name linear.[2] Totality also implies reflexivity, i.e.,a ≤ a. Therefore, a total order is also a partial order. The partial order has a weaker form of the third condition. (Itrequires only reflexivity, not totality.) An extension of a given partial order to a total order is called a linear extensionof that partial order.

69.1 Strict total order

For each (non-strict) total order ≤ there is an associated asymmetric (hence irreflexive) relation <, called a strict totalorder, which can equivalently be defined in two ways:

• a < b if and only if a ≤ b and a ≠ b

• a < b if and only if not b ≤ a (i.e., < is the inverse of the complement of ≤)

Properties:

• The relation is transitive: a < b and b < c implies a < c.• The relation is trichotomous: exactly one of a < b, b < a and a = b is true.• The relation is a strict weak order, where the associated equivalence is equality.

We can work the other way and start by choosing < as a transitive trichotomous binary relation; then a total order ≤can equivalently be defined in two ways:

• a ≤ b if and only if a < b or a = b

• a ≤ b if and only if not b < a

Two more associated orders are the complements ≥ and >, completing the quadruple {<, >, ≤, ≥}.We can define or explain the way a set is totally ordered by any of these four relations; the notation implies whetherwe are talking about the non-strict or the strict total order.

249



https://en.wikipedia.org/wiki/Infix_notation





















250 CHAPTER 69. TOTAL ORDER

69.2 Examples

• The letters of the alphabet ordered by the standard dictionary order, e.g., A < B < C etc.

• Any subset of a totally ordered set, with the restriction of the order on the whole set.

• Any set of cardinal numbers or ordinal numbers (more strongly, these are well-orders).

• If X is any set and f an injective function from X to a totally ordered set then f induces a total ordering on Xby setting x1 < x2 if and only if f(x1) < f(x2).

• The lexicographical order on the Cartesian product of a set of totally ordered sets indexed by an ordinal, isitself a total order. For example, any set of words ordered alphabetically is a totally ordered set, viewed as asubset of a Cartesian product of a countable number of copies of a set formed by adding the space symbol tothe alphabet (and defining a space to be less than any letter).

• The set of real numbers ordered by the usual less than (<) or greater than (>) relations is totally ordered, hencealso the subsets of natural numbers, integers, and rational numbers. Each of these can be shown to be the unique(to within isomorphism) smallest example of a totally ordered set with a certain property, (a total order A isthe smallest with a certain property if whenever B has the property, there is an order isomorphism from A to asubset of B):

• The natural numbers comprise the smallest totally ordered set with no upper bound.• The integers comprise the smallest totally ordered set with neither an upper nor a lower bound.• The rational numbers comprise the smallest totally ordered set which is dense in the real numbers. The

definition of density used here says that for every 'a' and 'b' in the real numbers such that 'a' < 'b', there isa 'q' in the rational numbers such that 'a' < 'q' < 'b'.

• The real numbers comprise the smallest unbounded totally ordered set that is connected in the ordertopology (defined below).

• Ordered fields are totally ordered by definition. They include the rational numbers and the real numbers.

69.3 Further concepts

69.3.1 Chains

While chain is sometimes merely a synonym for totally ordered set, it can also refer to a totally ordered subset ofsome partially ordered set. The latter definition has a crucial role in Zorn’s lemma.For example, consider the set of all subsets of the integers partially ordered by inclusion. Then the set { In : n is anatural number}, where In is the set of natural numbers below n, is a chain in this ordering, as it is totally orderedunder inclusion: If n≤k, then In is a subset of Ik.

69.3.2 Lattice theory

One may define a totally ordered set as a particular kind of lattice, namely one in which we have

{a ∨ b, a ∧ b} = {a, b} for all a, b.

We then write a ≤ b if and only if a = a ∧ b . Hence a totally ordered set is a distributive lattice.

https://en.wikipedia.org/wiki/Alphabetical_order








https://en.wikipedia.org/wiki/Countable_set

https://en.wikipedia.org/wiki/Real_numbers



https://en.wikipedia.org/wiki/Rational_numbers



https://en.wikipedia.org/wiki/Lower_bound

https://en.wikipedia.org/wiki/Dense_order

https://en.wikipedia.org/wiki/Connectedness

https://en.wikipedia.org/wiki/Ordered_field

https://en.wikipedia.org/wiki/Rational_numbers



https://en.wikipedia.org/wiki/Zorn%2527s_lemma


https://en.wikipedia.org/wiki/Partially_ordered





https://en.wikipedia.org/wiki/Distributive_lattice

69.3. FURTHER CONCEPTS 251

69.3.3 Finite total orders

A simple counting argument will verify that any non-empty finite totally ordered set (and hence any non-empty subsetthereof) has a least element. Thus every finite total order is in fact a well order. Either by direct proof or by observingthat every well order is order isomorphic to an ordinal one may show that every finite total order is order isomorphicto an initial segment of the natural numbers ordered by <. In other words a total order on a set with k elements inducesa bijection with the first k natural numbers. Hence it is common to index finite total orders or well orders with ordertype ω by natural numbers in a fashion which respects the ordering (either starting with zero or with one).

69.3.4 Category theory

Totally ordered sets form a full subcategory of the category of partially ordered sets, with the morphisms being mapswhich respect the orders, i.e. maps f such that if a ≤ b then f(a) ≤ f(b).A bijective map between two totally ordered sets that respects the two orders is an isomorphism in this category.

69.3.5 Order topology

For any totally ordered set X we can define the open intervals (a, b) = {x : a < x and x < b}, (−∞, b) = {x : x < b},(a, ∞) = {x : a < x} and (−∞, ∞) = X. We can use these open intervals to define a topology on any ordered set, theorder topology.When more than one order is being used on a set one talks about the order topology induced by a particular order.For instance if N is the natural numbers, < is less than and > greater than we might refer to the order topology onN induced by < and the order topology on N induced by > (in this case they happen to be identical but will not ingeneral).The order topology induced by a total order may be shown to be hereditarily normal.

69.3.6 Completeness

A totally ordered set is said to be complete if every nonempty subset that has an upper bound, has a least upperbound. For example, the set of real numbers R is complete but the set of rational numbers Q is not.There are a number of results relating properties of the order topology to the completeness of X:

• If the order topology on X is connected, X is complete.

• X is connected under the order topology if and only if it is complete and there is no gap in X (a gap is twopoints a and b in X with a < b such that no c satisfies a < c < b.)

• X is complete if and only if every bounded set that is closed in the order topology is compact.

A totally ordered set (with its order topology) which is a complete lattice is compact. Examples are the closed intervalsof real numbers, e.g. the unit interval [0,1], and the affinely extended real number system (extended real number line).There are order-preserving homeomorphisms between these examples.

69.3.7 Sums of orders

For any two disjoint total orders (A1,≤1) and (A2,≤2) , there is a natural order ≤+ on the set A1 ∪ A2 , which iscalled the sum of the two orders or sometimes just A1 +A2 :

For x, y ∈ A1 ∪A2 , x ≤+ y holds if and only if one of the following holds:

1. x, y ∈ A1 and x ≤1 y

2. x, y ∈ A2 and x ≤2 y

3. x ∈ A1 and y ∈ A2

https://en.wikipedia.org/wiki/Counting

https://en.wikipedia.org/wiki/Well_order

https://en.wikipedia.org/wiki/Order_isomorphic



https://en.wikipedia.org/wiki/Initial_segment

https://en.wikipedia.org/wiki/Order_type


https://en.wikipedia.org/wiki/Subcategory









https://en.wikipedia.org/wiki/Order_topology

https://en.wikipedia.org/wiki/Normal_space






https://en.wikipedia.org/wiki/Complete_lattice

https://en.wikipedia.org/wiki/Compact_space

https://en.wikipedia.org/wiki/Unit_interval

https://en.wikipedia.org/wiki/Affinely_extended_real_number_system


252 CHAPTER 69. TOTAL ORDER

Intutitively, this means that the elements of the second set are added on top of the elements of the first set.More generally, if (I,≤) is a totally ordered index set, and for each i ∈ I the structure (Ai,≤i) is a linear order,where the sets Ai are pairwise disjoint, then the natural total order on

∪iAi is defined by

For x, y ∈∪i∈I Ai , x ≤ y holds if:

1. Either there is some i ∈ I with x ≤i y2. or there are some i < j in I with x ∈ Ai , y ∈ Aj

69.4 Orders on the Cartesian product of totally ordered sets

In order of increasing strength, i.e., decreasing sets of pairs, three of the possible orders on the Cartesian product oftwo totally ordered sets are:

• Lexicographical order: (a,b) ≤ (c,d) if and only if a < c or (a = c and b ≤ d). This is a total order.• (a,b) ≤ (c,d) if and only if a ≤ c and b ≤ d (the product order). This is a partial order.• (a,b) ≤ (c,d) if and only if (a < c and b < d) or (a = c and b = d) (the reflexive closure of the direct product of

the corresponding strict total orders). This is also a partial order.

All three can similarly be defined for the Cartesian product of more than two sets.Applied to the vector space Rn, each of these make it an ordered vector space.See also examples of partially ordered sets.A real function of n real variables defined on a subset of Rn defines a strict weak order and a corresponding totalpreorder on that subset.

69.5 Related structures

A binary relation that is antisymmetric, transitive, and reflexive (but not necessarily total) is a partial order.A group with a compatible total order is a totally ordered group.There are only a few nontrivial structures that are (interdefinable as) reducts of a total order. Forgetting the orientationresults in a betweenness relation. Forgetting the location of the ends results in a cyclic order. Forgetting both dataresults in a separation relation.[3]

69.6 See also• Order theory• Well-order• Suslin’s problem• Countryman line

69.7 Notes[1] Nederpelt, Rob (2004). “Chapter 20.2: Ordered Sets. Orderings”. Logical Reasoning: A First Course. Texts in Computing

3 (3rd, Revised ed.). King’s College Publications. p. 325. ISBN 0-9543006-7-X.

[2] Nederpelt, Rob (2004). “Chapter 20.3: Ordered Sets. Linear orderings”. Logical Reasoning: A First Course. Texts inComputing 3 (3rd, Revisied ed.). King’s College Publications. p. 330. ISBN 0-9543006-7-X.

[3] Macpherson, H. Dugald (2011), “A survey of homogeneous structures” (PDF),DiscreteMathematics, doi:10.1016/j.disc.2011.01.024,retrieved 28 April 2011




https://en.wikipedia.org/wiki/Direct_product#Direct_product_of_binary_relations


https://en.wikipedia.org/wiki/Ordered_vector_space

https://en.wikipedia.org/wiki/Partially_ordered_set#Examples

https://en.wikipedia.org/wiki/Strict_weak_ordering#Function

https://en.wikipedia.org/wiki/Strict_weak_ordering#Function



https://en.wikipedia.org/wiki/Totally_ordered_group

https://en.wikipedia.org/wiki/Betweenness_relation

https://en.wikipedia.org/wiki/Cyclic_order

https://en.wikipedia.org/wiki/Separation_relation



https://en.wikipedia.org/wiki/Suslin%2527s_problem

https://en.wikipedia.org/wiki/Countryman_line





http://www.amsta.leeds.ac.uk/pure/staff/macpherson/homog_final2.pdf




69.8 References• George Grätzer (1971). Lattice theory: first concepts and distributive lattices. W. H. Freeman and Co. ISBN

0-7167-0442-0

• John G. Hocking and Gail S. Young (1961). Topology. Corrected reprint, Dover, 1988. ISBN 0-486-65676-4




Chapter 70

Total relation

In mathematics, a binary relation R over a set X is total or complete if for all a and b in X, a is related to b or b isrelated to a (or both).In mathematical notation, this is

∀a, b ∈ X, aRb ∨ bRa.

Total relations are sometimes said to have comparability.

70.1 Examples

For example, “is less than or equal to” is a total relation over the set of real numbers, because for two numbers eitherthe first is less than or equal to the second, or the second is less than or equal to the first. On the other hand, “is lessthan” is not a total relation, since one can pick two equal numbers, and then neither the first is less than the second, noris the second less than the first. (But note that “is less than” is a weak order which gives rise to a total order, namely“is less than or equal to”. The relationship between strict orders and weak orders is discussed at partially ordered set.)The relation “is a subset of” is also not total because, for example, neither of the sets {1,2} and {3,4} is a subset ofthe other.

70.2 Properties and related notions

Totality implies reflexivity.If a transitive relation is also total, it is a total preorder. If a partial order is also total, it is a total order.A binary relation R over X is called connex if for all a and b in X such that a ≠ b, a is related to b or b is related to a(or both):[1]

∀a, b ∈ X, aRb ∨ bRa ∨ (a = b).

Connexity does not imply reflexivity. A strict partial order is a strict total order if and only if it is connex.

70.3 See also

• Total order

254













70.4 References[1] Rautenberg, Wolfgang (2010), AConcise Introduction toMathematical Logic (3rd ed.), New York: Springer Science+Business

Media, doi:10.1007/978-1-4419-1221-3, ISBN 978-1-4419-1220-6

https://en.wikipedia.org/wiki/Wolfgang_Rautenberg

http://www.springerlink.com/content/978-1-4419-1220-6/




https://dx.doi.org/10.1007%252F978-1-4419-1221-3



Chapter 71

Transitive closure

For other uses, see Closure (disambiguation).This article is about the transitive closure of a binary relation. For the transitive closure of a set, see transitiveset#Transitive closure.

In mathematics, the transitive closure of a binary relation R on a set X is the transitive relation R+ on set X such thatR+ contains R and R+ is minimal (Lidl and Pilz 1998:337). If the binary relation itself is transitive, then the transitiveclosure is that same binary relation; otherwise, the transitive closure is a different relation. For example, if X is a setof airports and x R y means “there is a direct flight from airport x to airport y", then the transitive closure of R on Xis the relation R+: “it is possible to fly from x to y in one or more flights.”

71.1 Transitive relations and examples

A relation R on a set X is transitive if, for all x, y, z in X, whenever x R y and y R z then x R z. Examples of transitiverelations include the equality relation on any set, the “less than or equal” relation on any linearly ordered set, and therelation "x was born before y" on the set of all people. Symbolically, this can be denoted as: if x < y and y < z then x< z.One example of a non-transitive relation is “city x can be reached via a direct flight from city y" on the set of all cities.Simply because there is a direct flight from one city to a second city, and a direct flight from the second city to thethird, does not imply there is a direct flight from the first city to the third. The transitive closure of this relation is adifferent relation, namely “there is a sequence of direct flights that begins at city x and ends at city y". Every relationcan be extended in a similar way to a transitive relation.An example of a non-transitive relation with a less meaningful transitive closure is "x is the day of the week after y".The transitive closure of this relation is “some day x comes after a day y on the calendar”, which is trivially true for alldays of the week x and y (and thus equivalent to the Cartesian square, which is "x and y are both days of the week”).

71.2 Existence and description

For any relation R, the transitive closure of R always exists. To see this, note that the intersection of any family oftransitive relations is again transitive. Furthermore, there exists at least one transitive relation containing R, namelythe trivial one: X × X. The transitive closure of R is then given by the intersection of all transitive relations containingR.For finite sets, we can construct the transitive closure step by step, starting from R and adding transitive edges. Thisgives the intuition for a general construction. For any set X, we can prove that transitive closure is given by thefollowing expression

R+ =∪

i∈{1,2,3,...}

Ri.

256

https://en.wikipedia.org/wiki/Closure_(disambiguation)

https://en.wikipedia.org/wiki/Transitive_set#Transitive_closure

https://en.wikipedia.org/wiki/Transitive_set#Transitive_closure







https://en.wikipedia.org/wiki/Day_of_the_week



https://en.wikipedia.org/wiki/Indexed_family

https://en.wikipedia.org/wiki/There_exists

71.3. PROPERTIES 257

where Ri is the i-th power of R, defined inductively by

R1 = R

and, for i > 0 ,

Ri+1 = R ◦Ri

where ◦ denotes composition of relations.To show that the above definition of R+ is the least transitive relation containing R, we show that it contains R, that itis transitive, and that it is the smallest set with both of those characteristics.

• R ⊆ R+ : R+ contains all of the Ri , so in particular R+ contains R .

• R+ is transitive: every element ofR+ is in one of theRi , soR+ must be transitive by the following reasoning:if (s1, s2) ∈ Rj and (s2, s3) ∈ Rk , then from composition’s associativity, (s1, s3) ∈ Rj+k (and thus in R+

) because of the definition of Ri .

• R+ is minimal: LetG be any transitive relation containingR , we want to show thatR+ ⊆ G . It is sufficient toshow that for every i > 0 , Ri ⊆ G . Well, sinceG containsR , R1 ⊆ G . And sinceG is transitive, wheneverRi ⊆ G , Ri+1 ⊆ G according to the construction of Ri and what it means to be transitive. Therefore, byinduction, G contains every Ri , and thus also R+ .

71.3 Properties

The intersection of two transitive relations is transitive.The union of two transitive relations need not be transitive. To preserve transitivity, one must take the transitiveclosure. This occurs, for example, when taking the union of two equivalence relations or two preorders. To obtain anew equivalence relation or preorder one must take the transitive closure (reflexivity and symmetry—in the case ofequivalence relations—are automatic).

71.4 In graph theory

In computer science, the concept of transitive closure can be thought of as constructing a data structure that makesit possible to answer reachability questions. That is, can one get from node a to node d in one or more hops? Abinary relation tells you only that node a is connected to node b, and that node b is connected to node c, etc. Afterthe transitive closure is constructed, as depicted in the following figure, in an O(1) operation one may determine thatnode d is reachable from node a. The data structure is typically stored as a matrix, so if matrix[1][4] = 1, then it isthe case that node 1 can reach node 4 through one or more hops.The transitive closure of a directed acyclic graph (DAG) is the reachability relation of the DAG and a strict partialorder.

71.5 In logic and computational complexity

The transitive closure of a binary relation cannot, in general, be expressed in first-order logic (FO). This means thatone cannot write a formula using predicate symbols R and T that will be satisfied in any model if and only if T isthe transitive closure of R. In finite model theory, first-order logic (FO) extended with a transitive closure operatoris usually called transitive closure logic, and abbreviated FO(TC) or just TC. TC is a sub-type of fixpoint logics.The fact that FO(TC) is strictly more expressive than FO was discovered by Ronald Fagin in 1974; the result wasthen rediscovered by Alfred Aho and Jeffrey Ullman in 1979, who proposed to use fixpoint logic as a database query












https://en.wikipedia.org/wiki/Finite_model_theory

https://en.wikipedia.org/wiki/Fixpoint_logic

https://en.wikipedia.org/wiki/Ronald_Fagin

https://en.wikipedia.org/wiki/Alfred_Aho

https://en.wikipedia.org/wiki/Jeffrey_Ullman

https://en.wikipedia.org/wiki/Database_query_language

258 CHAPTER 71. TRANSITIVE CLOSURE

Output

Input

Transitive closure constructs the output graph from the input graph.

language (Libkin 2004:vii). With more recent concepts of finite model theory, proof that FO(TC) is strictly moreexpressive than FO follows immediately from the fact that FO(TC) is not Gaifman-local (Libkin 2004:49).In computational complexity theory, the complexity class NL corresponds precisely to the set of logical sentencesexpressible in TC. This is because the transitive closure property has a close relationship with the NL-completeproblem STCON for finding directed paths in a graph. Similarly, the class L is first-order logic with the commutative,transitive closure. When transitive closure is added to second-order logic instead, we obtain PSPACE.

71.6 In database query languages

Further information: Hierarchical and recursive queries in SQL

Since the 1980s Oracle Database has implemented a proprietary SQL extension CONNECT BY... START WITHthat allows the computation of a transitive closure as part of a declarative query. The SQL 3 (1999) standard addeda more general WITH RECURSIVE construct also allowing transitive closures to be computed inside the queryprocessor; as of 2011 the latter is implemented in IBM DB2, Microsoft SQL Server, and PostgreSQL, although notin MySQL (Benedikt and Senellart 2011:189).Datalog also implements transitive closure computations (Silberschatz et al. 2010:C.3.6).

71.7 Algorithms

Efficient algorithms for computing the transitive closure of a graph can be found in Nuutila (1995). The fastest worst-case methods, which are not practical, reduce the problem to matrix multiplication. The problem can also be solvedby the Floyd–Warshall algorithm, or by repeated breadth-first search or depth-first search starting from each node ofthe graph.More recent research has explored efficient ways of computing transitive closure on distributed systems based on theMapReduce paradigm (Afrati et al. 2011).

71.8 See also• Deductive closure




https://en.wikipedia.org/wiki/Gaifman-local

https://en.wikipedia.org/wiki/Computational_complexity_theory

https://en.wikipedia.org/wiki/Complexity_class

https://en.wikipedia.org/wiki/NL_(complexity)

https://en.wikipedia.org/wiki/NL-complete

https://en.wikipedia.org/wiki/STCON

https://en.wikipedia.org/wiki/Directed_path

https://en.wikipedia.org/wiki/L_(complexity)

https://en.wikipedia.org/wiki/Second-order_logic

https://en.wikipedia.org/wiki/PSPACE

https://en.wikipedia.org/wiki/Hierarchical_and_recursive_queries_in_SQL

https://en.wikipedia.org/wiki/Oracle_Database

https://en.wikipedia.org/wiki/SQL

https://en.wikipedia.org/wiki/SQL_3

https://en.wikipedia.org/wiki/IBM_DB2

https://en.wikipedia.org/wiki/Microsoft_SQL_Server

https://en.wikipedia.org/wiki/PostgreSQL

https://en.wikipedia.org/wiki/MySQL

https://en.wikipedia.org/wiki/Datalog

https://en.wikipedia.org/wiki/Matrix_multiplication

https://en.wikipedia.org/wiki/Floyd%E2%80%93Warshall_algorithm

https://en.wikipedia.org/wiki/Breadth-first_search

https://en.wikipedia.org/wiki/Depth-first_search

https://en.wikipedia.org/wiki/MapReduce

https://en.wikipedia.org/wiki/Deductive_closure


• Transitive reduction (a smallest relation having the transitive closure of R as its transitive closure)

• Symmetric closure

• Reflexive closure

• Ancestral relation

71.9 References• Lidl, R. and Pilz, G., 1998, Applied abstract algebra, 2nd edition, Undergraduate Texts in Mathematics,

Springer, ISBN 0-387-98290-6

• Keller, U., 2004, Some Remarks on the Definability of Transitive Closure in First-order Logic and Datalog(unpublished manuscript)

• Erich Grädel; Phokion G. Kolaitis; Leonid Libkin; Maarten Marx; Joel Spencer; Moshe Y. Vardi; Yde Venema;Scott Weinstein (2007). Finite Model Theory and Its Applications. Springer. pp. 151–152. ISBN 978-3-540-68804-4.

• Libkin, Leonid (2004), Elements of Finite Model Theory, Springer, ISBN 978-3-540-21202-7

• Heinz-Dieter Ebbinghaus; Jörg Flum (1999). FiniteModel Theory (2nd ed.). Springer. pp. 123–124, 151–161,220–235. ISBN 978-3-540-28787-2.

• Aho, A. V.; Ullman, J. D. (1979). “Universality of data retrieval languages”. Proceedings of the 6th ACMSIGACT-SIGPLAN Symposium on Principles of programming languages - POPL '79. p. 110. doi:10.1145/567752.567763.

• Benedikt, M.; Senellart, P. (2011). “Databases”. In Blum, Edward K.; Aho, Alfred V. Computer Science. TheHardware, Software and Heart of It. pp. 169–229. doi:10.1007/978-1-4614-1168-0_10. ISBN 978-1-4614-1167-3.

• Nuutila, E., Efficient Transitive Closure Computation in Large Digraphs. Acta Polytechnica Scandinavica,Mathematics and Computing in Engineering Series No. 74, Helsinki 1995, 124 pages. Published by theFinnish Academy of Technology. ISBN 951-666-451-2, ISSN 1237-2404, UDC 681.3.

• Abraham Silberschatz; Henry Korth; S. Sudarshan (2010). Database System Concepts (6th ed.). McGraw-Hill.ISBN 978-0-07-352332-3. Appendix C (online only)

• Foto N. Afrati, Vinayak Borkar, Michael Carey, Neoklis Polyzotis, Jeffrey D. Ullman, Map-Reduce Extensionsand Recursive Queries, EDBT 2011, March 22–24, 2011, Uppsala, Sweden, ISBN 978-1-4503-0528-0

71.10 External links• "Transitive closure and reduction", The Stony Brook Algorithm Repository, Steven Skiena .

• "Apti Algoritmi", An example and some C++ implementations of algorithms that calculate the transitive closureof a given binary relation, Vreda Pieterse.


https://en.wikipedia.org/wiki/Symmetric_closure


https://en.wikipedia.org/wiki/Ancestral_relation

https://en.wikipedia.org/wiki/Undergraduate_Texts_in_Mathematics






https://en.wikipedia.org/wiki/Leonid_Libkin






https://dx.doi.org/10.1145%252F567752.567763


https://dx.doi.org/10.1007%252F978-1-4614-1168-0_10




http://www.cs.hut.fi/~enu/thesis.html




http://codex.cs.yale.edu/avi/db-book/db6/appendices-dir/c.pdf

http://www.edbt.org/Proceedings/2011-Uppsala/papers/edbt/a1-afrati.pdf

http://www.edbt.org/Proceedings/2011-Uppsala/papers/edbt/a1-afrati.pdf


http://www.cs.sunysb.edu/~algorith/files/transitive-closure.shtml

http://www.cs.up.ac.za/cs/vpieterse/AptiAlgo/AptiAlgoritmi.html

Chapter 72

Transitive relation

In mathematics, a binary relation R over a set X is transitive if whenever an element a is related to an element b,and b is in turn related to an element c, then a is also related to c. Transitivity is a key property of both partial orderrelations and equivalence relations.


In terms of set theory, the transitive relation can be defined as:

∀a, b, c ∈ X : (aRb ∧ bRc) ⇒ aRc

72.2 Examples

For example, “is greater than,” “is at least as great as,” and “is equal to” (equality) are transitive relations:

whenever A > B and B > C, then also A > Cwhenever A ≥ B and B ≥ C, then also A ≥ Cwhenever A = B and B = C, then also A = C.

On the other hand, “is the mother of” is not a transitive relation, because if Alice is the mother of Brenda, and Brendais the mother of Claire, then Alice is not the mother of Claire. What is more, it is antitransitive: Alice can never bethe mother of Claire.Then again, in biology we often need to consider motherhood over an arbitrary number of generations: the relation“is a matrilinear ancestor of”. This is a transitive relation. More precisely, it is the transitive closure of the relation“is the mother of”.More examples of transitive relations:

• “is a subset of” (set inclusion)

• “divides” (divisibility)

• “implies” (implication)

72.3 Properties

260








https://en.wikipedia.org/wiki/Antitransitive

https://en.wikipedia.org/wiki/Matrilinear



https://en.wikipedia.org/wiki/Set_inclusion



72.4. COUNTING TRANSITIVE RELATIONS 261

72.3.1 Closure properties

The converse of a transitive relation is always transitive: e.g. knowing that “is a subset of” is transitive and “is asuperset of” is its converse, we can conclude that the latter is transitive as well.The intersection of two transitive relations is always transitive: knowing that “was born before” and “has the same firstname as” are transitive, we can conclude that “was born before and also has the same first name as” is also transitive.The union of two transitive relations is not always transitive. For instance “was born before or has the same first nameas” is not generally a transitive relation.The complement of a transitive relation is not always transitive. For instance, while “equal to” is transitive, “not equalto” is only transitive on sets with at most one element.

72.3.2 Other properties

A transitive relation is asymmetric if and only if it is irreflexive.[1]

72.3.3 Properties that require transitivity

• Preorder – a reflexive transitive relation

• partial order – an antisymmetric preorder

• Total preorder – a total preorder

• Equivalence relation – a symmetric preorder

• Strict weak ordering – a strict partial order in which incomparability is an equivalence relation

• Total ordering – a total, antisymmetric transitive relation

72.4 Counting transitive relations

No general formula that counts the number of transitive relations on a finite set (sequence A006905 in OEIS) isknown.[2] However, there is a formula for finding the number of relations that are simultaneously reflexive, symmet-ric, and transitive – in other words, equivalence relations – (sequence A000110 in OEIS), those that are symmetricand transitive, those that are symmetric, transitive, and antisymmetric, and those that are total, transitive, and an-tisymmetric. Pfeiffer[3] has made some progress in this direction, expressing relations with combinations of theseproperties in terms of each other, but still calculating any one is difficult. See also.[4]

72.5 See also

• Transitive closure

• Transitive reduction

• Intransitivity


• Symmetric relation

• Quasitransitive relation

• Nontransitive dice

• Rational choice theory









https://en.wikipedia.org/wiki/Total_preorder





https://en.wikipedia.org/wiki/Total_ordering













https://en.wikipedia.org/wiki/Quasitransitive_relation

https://en.wikipedia.org/wiki/Nontransitive_dice

https://en.wikipedia.org/wiki/Rational_choice_theory#Formal_statement

262 CHAPTER 72. TRANSITIVE RELATION

72.6 Sources

72.6.1 References[1] Flaška, V.; Ježek, J.; Kepka, T.; Kortelainen, J. (2007). Transitive Closures of Binary Relations I (PDF). Prague: School

of Mathematics - Physics Charles University. p. 1. Lemma 1.1 (iv). Note that this source refers to asymmetric relationsas “strictly antisymmetric”.

[2] Steven R. Finch, “Transitive relations, topologies and partial orders”, 2003.

[3] Götz Pfeiffer, "Counting Transitive Relations", Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.

[4] Gunnar Brinkmann and Brendan D. McKay,”Counting unlabelled topologies and transitive relations"

72.6.2 Bibliography

• Ralph P. Grimaldi, Discrete and Combinatorial Mathematics, ISBN 0-201-19912-2.


72.7 External links• Hazewinkel, Michiel, ed. (2001), “Transitivity”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-

010-4

• Transitivity in Action at cut-the-knot


http://algo.inria.fr/csolve/posets.pdf

http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Pfeiffer/pfeiffer6.html

http://cs.anu.edu.au/~bdm/papers/topologies.pdf

https://en.wikipedia.org/wiki/Ralph_Grimaldi




http://www.encyclopediaofmath.org/index.php?title=p/t093810






http://www.cut-the-knot.org/triangle/remarkable.shtml

https://en.wikipedia.org/wiki/Cut-the-knot

Chapter 73

Trichotomy (mathematics)

In mathematics, the Law of Trichotomy states that every real number is either positive, negative, or zero.[1] Moregenerally, trichotomy is the property of an order relation < on a set X that for any x and y, exactly one of the followingholds: x < y , x = y , or x > y .In mathematical notation, this is

∀x ∈ X ∀y ∈ X ((x < y ∧¬(y < x)∧¬(x = y) )∨ (¬(x < y)∧ y < x∧¬(x = y) )∨ (¬(x < y)∧¬(y < x)∧x = y )) .

Assuming that the ordering is irreflexive and transitive, this can be simplified to

∀x ∈ X ∀y ∈ X ((x < y) ∨ (y < x) ∨ (x = y)) .

In classical logic, this axiom of trichotomy holds for ordinary comparison between real numbers and therefore alsofor comparisons between integers and between rational numbers. The law does not hold in general in intuitionisticlogic.In ZF set theory and Bernays set theory, the law of trichotomy holds between the cardinal numbers of well-orderablesets even without the axiom of choice. If the axiom of choice holds, then trichotomy holds between arbitrary cardinalnumbers (because they are all well-orderable in that case).[2]

More generally, a binary relation R on X is trichotomous if for all x and y in X exactly one of xRy, yRx or x=y holds.If such a relation is also transitive it is a strict total order; this is a special case of a strict weak order. For example,in the case of three element set {a,b,c} the relation R given by aRb, aRc, bRc is a strict total order, while the relationR given by the cyclic aRb, bRc, cRa is a non-transitive trichotomous relation.In the definition of an ordered integral domain or ordered field, the law of trichotomy is usually taken as morefoundational than the law of total order.A trichotomous relation cannot be reflexive, since xRx must be false. If a trichotomous relation is transitive, it istrivially antisymmetric and also asymmetric, since xRy and yRx cannot both hold.

73.1 See also• Dichotomy

• Law of noncontradiction

• Law of excluded middle

73.2 References[1] http://mathworld.wolfram.com/TrichotomyLaw.html

263








https://en.wikipedia.org/wiki/Intuitionistic_logic

https://en.wikipedia.org/wiki/Intuitionistic_logic







https://en.wikipedia.org/wiki/Total_order#Strict_total_order


https://en.wikipedia.org/wiki/Ordered_integral_domain

https://en.wikipedia.org/wiki/Ordered_field





https://en.wikipedia.org/wiki/Dichotomy

https://en.wikipedia.org/wiki/Law_of_noncontradiction

https://en.wikipedia.org/wiki/Law_of_excluded_middle

http://mathworld.wolfram.com/TrichotomyLaw.html

264 CHAPTER 73. TRICHOTOMY (MATHEMATICS)

[2] Bernays, Paul (1991). Axiomatic Set Theory. Dover Publications. ISBN 0-486-66637-9.



Chapter 74

Unimodality

“Unimodal” redirects here. For the company that promotes personal rapid transit, see SkyTran.

In mathematics, unimodality means possessing a unique mode. More generally, unimodality means there is only asingle highest value, somehow defined, of some mathematical object.[1]

74.1 Unimodal probability distribution

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-5 -4 -3 -2 -1 0 1 2 3 4 5

μ = 0, σ² = 0.2μ = 0, σ² = 1.0μ = 0, σ² = 5.0

μ = -2, σ² = 0.5

Figure 1. probability density function of normal distributions, an example of unimodal distribution.

In statistics, a unimodal probability distribution (or when referring to the distribution, a unimodal distribution)

265

https://en.wikipedia.org/wiki/SkyTran


https://en.wikipedia.org/wiki/Mode_(statistics)

https://en.wikipedia.org/wiki/Probability_density_function

https://en.wikipedia.org/wiki/Statistics

266 CHAPTER 74. UNIMODALITY

Figure 2. a simple bimodal distribution.

Figure 3. a distribution which, though strictly unimodal, is usually referred to as bimodal.

is a probability distribution which has a single mode. As the term “mode” has multiple meanings, so does the term“unimodal”.Strictly speaking, a mode of a discrete probability distribution is a value at which the probability mass function (pmf)takes its maximum value. In other words, it is a most likely value. A mode of a continuous probability distributionis a value at which the probability density function (pdf) attains its maximum value. Note that in both cases therecan be more than one mode, since the maximum value of either the pmf or the pdf can be attained at more than onevalue.If there is a single mode, the distribution function is called “unimodal”. If it has more modes it is “bimodal” (2), “tri-modal” (3), etc., or in general, “multimodal”.[2] Figure 1 illustrates normal distributions, which are unimodal. Otherexamples of unimodal distributions include Cauchy distribution, Student’s t-distribution and chi-squared distribution.

https://en.wikipedia.org/wiki/Probability_distribution

https://en.wikipedia.org/wiki/Discrete_probability_distribution

https://en.wikipedia.org/wiki/Probability_mass_function

https://en.wikipedia.org/wiki/Continuous_probability_distribution

https://en.wikipedia.org/wiki/Probability_density_function

https://en.wikipedia.org/wiki/Normal_distribution

https://en.wikipedia.org/wiki/Cauchy_distribution

https://en.wikipedia.org/wiki/Student%2527s_t-distribution

https://en.wikipedia.org/wiki/Chi-squared_distribution

74.1. UNIMODAL PROBABILITY DISTRIBUTION 267

Figure 2 illustrates a bimodal distribution.Figure 3 illustrates a distribution which by strict definition is unimodal. However, confusingly, and mostly withcontinuous distributions, when a pdf function has multiple local maxima it is common to refer to all of the localmaxima as modes of the distribution. Therefore, if a pdf has more than one local maximum it is referred to asmultimodal. Under this common definition, Figure 3 illustrates a bimodal distribution.

74.1.1 Other definitions

Other definitions of unimodality in distribution functions also exist.In continuous distributions, unimodality can be defined through the behavior of the cumulative distribution function(cdf).[3] If the cdf is convex for x < m and concave for x > m, then the distribution is unimodal, m being the mode.Note that under this definition the uniform distribution is unimodal,[4] as well as any other distribution in which themaximum distribution is achieved for a range of values, e.g. trapezoidal distribution. Note also that usually thisdefinition allows for a discontinuity at the mode; usually in a continuous distribution the probability of any singlevalue is zero, while this definition allows for a non-zero probability, or an “atom of probability”, at the mode.Criteria for unimodality can also be defined through the characteristic function of the distribution[3] or through itsLaplace–Stieltjes transform.[5]

Another way to define a unimodal discrete distribution is by the occurrence of sign changes in the sequence of dif-ferences of the probabilities.[6] A discrete distribution with a probability mass function, {pn;n = . . . ,−1, 0, 1, . . . }, is called unimodal if the sequence . . . , p−2 − p−1, p−1 − p0, p0 − p1, p1 − p2, . . . has exactly one sign change(when zeroes don't count).

74.1.2 Uses and results

One reason for the importance of distribution unimodality is that it allows for several important results. SeveralInequalities are given below which are only valid for unimodal distributions. Thus, it is important to assess whetheror not a given data set comes from a unimodal distribution. Several tests for unimodality are given in the Wiki articleon Multimodal Distribution

74.1.3 Gauss’ inequality

A first important result is Gauss’s inequality.[7] Gauss’s inequality gives an upper bound on the probability that a valuelies more than any given distance from its mode. This inequality depends on unimodality.

74.1.4 Vysochanskiï–Petunin inequality

A second is the Vysochanskiï–Petunin inequality,[8] a refinement of the Chebyshev inequality. The Chebyshev in-equality guarantees that in any probability distribution, “nearly all” the values are “close to” the mean value. TheVysochanskiï–Petunin inequality refines this to even nearer values, provided that the distribution function is uni-modal. Further results were shown by Sellke & Sellke.[9]

74.1.5 Mode, median and mean

It can be shown for a unimodal distribution that the median X̃ and the mean X̄ lie within (3/5)1/2 ≈ 0.7746 standarddeviations of each other.[10] In symbols,

∣∣∣X̃ − X̄∣∣∣

σ≤ (3/5)1/2

where |.| is the absolute value.A similar relation holds between the median and the mode: they lie within 31/2 ≈ 1.732 standard deviations of eachother:

https://en.wikipedia.org/wiki/Local_maximum

https://en.wikipedia.org/wiki/Cumulative_distribution_function

https://en.wikipedia.org/wiki/Convex_function

https://en.wikipedia.org/wiki/Concave_function

https://en.wikipedia.org/wiki/Uniform_distribution_(continuous)

https://en.wikipedia.org/wiki/Characteristic_function_(probability_theory)

https://en.wikipedia.org/wiki/Laplace%E2%80%93Stieltjes_transform

https://en.wikipedia.org/wiki/Probability_mass_function

https://en.wikipedia.org/wiki/Multimodal_distribution

https://en.wikipedia.org/wiki/Gauss%2527s_inequality

https://en.wikipedia.org/wiki/Vysochanski%C3%AF%E2%80%93Petunin_inequality

https://en.wikipedia.org/wiki/Chebyshev_inequality

268 CHAPTER 74. UNIMODALITY

∣∣∣X̃ − mode∣∣∣

σ≤ 31/2.

74.1.6 Skewness and kurtosis

Rohatgi and Szekely have shown that the skewness and kurtosis of a unimodal distribution are related by the inequality:[11]

γ2 − κ ≤ 6

5

where κ is the kurtosis and γ is the skewness.Klaassen, Mokveld, and van Es derived a slightly different inequality (shown below) from the one derived by Rohatgiand Szekely (shown above), which tends to be more inclusive (i.e., yield more positives) in tests of unimodality:[12]

γ2 − κ ≤ 186

125

74.2 Unimodal function

As the term “modal” applies to data sets and probability distribution, and not in general to functions, the definitionsabove do not apply. The definition of “unimodal” was extended to functions of real numbers as well.A common definition is as follows: a function f(x) is a unimodal function if for some value m, it is monotonicallyincreasing for x ≤ m and monotonically decreasing for x ≥ m. In that case, the maximum value of f(x) is f(m) andthere are no other local maxima.Proving unimodality is often hard. One way consists in using the definition of that property, but it turns out to besuitable for simple functions only. A general method based on derivatives exists,[13] but it does not succeed for everyfunction despite its simplicity.Examples of unimodal functions include quadratic polynomial functions with a negative quadratic coefficient, tentmap functions, and more.The above is sometimes related to as “strong unimodality”, from the fact that the monotonicity implied is strongmonotonicity. A function f(x) is a weakly unimodal function if there exists a value m for which it is weakly mono-tonically increasing for x ≤ m and weakly monotonically decreasing for x ≥ m. In that case, the maximum valuef(m) can be reached for a continuous range of values of x. An example of a weakly unimodal function which is notstrongly unimodal is every other row in a Pascal triangle.Depending on context, unimodal function may also refer to a function that has only one local minimum, rather thanmaximum.[14] For example, local unimodal sampling, a method for doing numerical optimization, is often demon-strated with such a function. It can be said that a unimodal function under this extension is a function with a singlelocal extremum.One important property of unimodal functions is that the extremum can be found using search algorithms such asgolden section search, ternary search or successive parabolic interpolation.

74.3 Other extensions

A function f(x) is “S-unimodal” (often referred to as “S-unimodal map”) if its Schwarzian derivative is negative forall x ̸= c , where c is the critical point.[15]

In computational geometry if a function is unimodal it permits the design of efficient algorithms for finding theextrema of the function.[16]

https://en.wikipedia.org/wiki/Skewness

https://en.wikipedia.org/wiki/Kurtosis



https://en.wikipedia.org/wiki/Monotonic

https://en.wikipedia.org/wiki/Maximum

https://en.wikipedia.org/wiki/Quadratic_polynomial

https://en.wikipedia.org/wiki/Tent_map

https://en.wikipedia.org/wiki/Tent_map

https://en.wikipedia.org/wiki/Pascal_triangle

https://en.wikipedia.org/wiki/Local_unimodal_sampling

https://en.wikipedia.org/wiki/Extremum

https://en.wikipedia.org/wiki/Search_algorithm

https://en.wikipedia.org/wiki/Golden_section_search

https://en.wikipedia.org/wiki/Ternary_search

https://en.wikipedia.org/wiki/Successive_parabolic_interpolation

https://en.wikipedia.org/wiki/Schwarzian_derivative

https://en.wikipedia.org/wiki/Computational_geometry

74.4. SEE ALSO 269

A more general definition, applicable to a function f(X) of a vector variable X is that f is unimodal if there is a one toone differentiable mapping X = G(Z) such that f(G(Z)) is convex. Usually one would want G(Z) to be continuouslydifferentiable with nonsingular Jacobian matrix.Quasiconvex functions and quasiconcave functions extend the concept of unimodality to functions whose argumentsbelong to higher-dimensional Euclidean spaces.

74.4 See also• Bimodal distribution

74.5 References[1] Weisstein, Eric W., “Unimodal”, MathWorld.

[2] Weisstein, Eric W., “Mode”, MathWorld.

[3] A.Ya. Khinchin (1938). “On unimodal distributions”. Trams. Res. Inst. Math. Mech. (in Russian) (University of Tomsk)2 (2): 1–7.

[4] Ushakov, N.G. (2001), “Unimodal distribution”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN978-1-55608-010-4

[5] Vladimirovich Gnedenko and Victor Yu Korolev (1996). Random summation: limit theorems and applications. CRC-Press.ISBN 0-8493-2875-6. p. 31

[6] Medgyessy, P. (March 1972). “On the unimodality of discrete distributions”. Periodica Mathematica Hungarica 2 (1–4):245–257. doi:10.1007/bf02018665.

[7] Gauss, C. F. (1823). “Theoria Combinationis Observationum Erroribus Minimis Obnoxiae, Pars Prior”. CommentationesSocietatis Regiae Scientiarum Gottingensis Recentiores 5.

[8] D. F. Vysochanskij, Y. I. Petunin (1980). “Justification of the 3σ rule for unimodal distributions”. Theory of Probabilityand Mathematical Statistics 21: 25–36.

[9] Sellke, T.M.; Sellke, S.H. (1997). “Chebyshev inequalities for unimodal distributions”. American Statistician (AmericanStatistical Association) 51 (1): 34–40. doi:10.2307/2684690. JSTOR 2684690.

[10] Basu, Sanjib, and Anirban DasGupta. “The mean, median, and mode of unimodal distributions: a characterization.” Theoryof Probability & Its Applications 41.2 (1997): 210-223.

[11] Rohatgi VK, Szekely GJ (1989) Sharp inequalities between skewness and kurtosis. Statistics & Probability Letters 8:297-299

[12] Klaassen CAJ, Mokveld PJ, van Es B (2000) Squared skewness minus kurtosis bounded by 186/125 for unimodal distri-butions. Stat & Prob Lett 50 (2) 131–135

[13] “On the unimodality of METRIC Approximation subject to normally distributed demands.” (PDF). Method in appendixD, Example in theorem 2 page 5. Retrieved 2013-08-28.

[14] “Mathematical Programming Glossary.”. Retrieved 2010-07-07.

[15] See e.g. John Guckenheimer and Stewart Johnson (July 1990). “Distortion of S-Unimodal Maps”. The Annals of Mathe-matics, Second Series 132 (1). pp. 71–130. doi:10.2307/1971501.

[16] Godfried T. Toussaint (June 1984). “Complexity, convexity, and unimodality”. International Journal of Computer andInformation Sciences 13 (3). pp. 197–217. doi:10.1007/bf00979872.

https://en.wikipedia.org/wiki/Quasiconvex_function


https://en.wikipedia.org/wiki/Bimodal_distribution


http://mathworld.wolfram.com/Unimodal.html



http://mathworld.wolfram.com/Mode.html


http://www.encyclopediaofmath.org/index.php?title=U/u095330







http://www.akademiai.com/content/j5012306777g764n/




https://en.wikipedia.org/wiki/American_Statistician


https://dx.doi.org/10.2307%252F2684690



http://epubs.siam.org/doi/pdf/10.1137/S0040585X97975447

http://epubs.siam.org/doi/pdf/10.1137/S0040585X97975447

http://homepage.univie.ac.at/thibaut.barthelemy/METRIC.pdf

http://glossary.computing.society.informs.org/second.php?page=U.html


https://dx.doi.org/10.2307%252F1971501



Chapter 75

Weak ordering

Not to be confused with weak order of permutations.In mathematics, especially order theory, a weak ordering is a mathematical formalization of the intuitive notion of

a<bc<b

a<ba<c

b<ac<a

b<ab<c

c<ac<b

a<cb<ca,b,c

c<b<a

b<c<a

b<a<c

c<a<b a<b<c

a<c<b

The 13 possible strict weak orderings on a set of three elements {a, b, c}. The only partially ordered sets are coloured, while totallyordered ones are in black. Two orderings are shown as connected by an edge if they differ by a single dichotomy.

270

https://en.wikipedia.org/wiki/Weak_order_of_permutations






75.1. EXAMPLES 271

a ranking of a set, some of whose members may be tied with each other. Weak orders are a generalization of totallyordered sets (rankings without ties) and are in turn generalized by partially ordered sets and preorders.[1]

There are several common ways of formalizing weak orderings, that are different from each other but cryptomorphic(interconvertable with no loss of information): they may be axiomatized as strict weak orderings (partially orderedsets in which incomparability is a transitive relation), as total preorders (transitive binary relations in which at leastone of the two possible relations exists between every pair of elements), or as ordered partitions (partitions of theelements into disjoint subsets, together with a total order on the subsets). In many cases another representation calleda preferential arrangement based on a utility function is also possible.Weak orderings are counted by the ordered Bell numbers. They are used in computer science as part of partitionrefinement algorithms, and in the C++ Standard Library.

75.1 Examples

In horse racing, the use of photo finishes has eliminated some, but not all, ties or (as they are called in this context)dead heats, so the outcome of a horse race may be modeled by a weak ordering.[2] In an example from the MarylandHunt Cup steeplechase in 2007, The Bruce was the clear winner, but two horses Bug River and Lear Charm tied forsecond place, with the remaining horses farther back; three horses did not finish.[3] In the weak ordering describingthis outcome, The Bruce would be first, Bug River and Lear Charm would be ranked after The Bruce but before allthe other horses that finished, and the three horses that did not finish would be placed last in the order but tied witheach other.The points of the Euclidean plane may be ordered by their distance from the origin, giving another example of a weakordering with infinitely many elements, infinitely many subsets of tied elements (the sets of points that belong to acommon circle centered at the origin), and infinitely many points within these subsets. Although this ordering has asmallest element (the origin itself), it does not have any second-smallest elements, nor any largest element.Opinion polling in political elections provides an example of a type of ordering that resembles weak orderings, but isbetter modeled mathematically in other ways. In the results of a poll, one candidate may be clearly ahead of another,or the two candidates may be statistically tied, meaning not that their poll results are equal but rather that they arewithin the margin of error of each other. However, if candidate x is statistically tied with y, and y is statistically tiedwith z, it might still be possible for x to be clearly better than z, so being tied is not in this case a transitive relation.Because of this possibility, rankings of this type are better modeled as semiorders than as weak orderings.[4]

75.2 Axiomatizations

75.2.1 Strict weak orderings

A strict weak ordering is a binary relation < on a set S that is a strict partial order (a transitive relation that isirreflexive, or equivalently,[5] that is asymmetric) in which the relation “neither a < b nor b < a" is transitive.[1]

The equivalence classes of this “incomparability relation” partition the elements of S, and are totally ordered by <.Conversely, any total order on a partition of S gives rise to a strict weak ordering in which x < y if and only if thereexists sets A and B in the partition with x in A, y in B, and A < B in the total order.As a non-example, consider the partial order in the set {a, b, c} defined by the relationship b < c. The pairs a,b and a,care incomparable but b and c are related, so incomparability does not form an equivalence relation and this exampleis not a strict weak ordering.A strict weak ordering has the following properties. For all x and y in S,

• For all x, it is not the case that x < x (irreflexivity).

• For all x, y, if x < y then it is not the case that y < x (asymmetry).

• For all x, y, and z, if x < y and y < z then x < z (transitivity).

• For all x, y, and z, if x is incomparable with y, and y is incomparable with z, then x is incomparable with z(transitivity of incomparability).

https://en.wikipedia.org/wiki/Ranking


https://en.wikipedia.org/wiki/Tie_(draw)





https://en.wikipedia.org/wiki/Cryptomorphism



https://en.wikipedia.org/wiki/Utility_function

https://en.wikipedia.org/wiki/Ordered_Bell_number


https://en.wikipedia.org/wiki/Partition_refinement


https://en.wikipedia.org/wiki/C++_Standard_Library

https://en.wikipedia.org/wiki/Horse_racing

https://en.wikipedia.org/wiki/Photo_finish

https://en.wikipedia.org/wiki/List_of_dead_heat_horse_races

https://en.wikipedia.org/wiki/Maryland_Hunt_Cup

https://en.wikipedia.org/wiki/Maryland_Hunt_Cup


https://en.wikipedia.org/wiki/Euclidean_distance

https://en.wikipedia.org/wiki/Origin_(mathematics)

https://en.wikipedia.org/wiki/Circle

https://en.wikipedia.org/wiki/Opinion_poll











https://en.wikipedia.org/wiki/Partition_(set_theory)




272 CHAPTER 75. WEAK ORDERING

This list of properties is somewhat redundant, as asymmetry follows readily from irreflexivity and transitivity.Transitivity of incomparability (together with transitivity) can also be stated in the following forms:

• If x < y, then for all z, either x < z or z < y or both.

Or:

• If x is incomparable with y, then for all z ≠ x, z ≠ y, either (x < z and y < z) or (z < x and z < y) or (z isincomparable with x and z is incomparable with y).

75.2.2 Total preorders

Strict weak orders are very closely related to total preorders or (non-strict) weak orders, and the same mathematicalconcepts that can be modeled with strict weak orderings can be modeled equally well with total preorders. A totalpreorder or weak order is a preorder that is total; that is, no pair of items is incomparable. A total preorder ≲ satisfiesthe following properties:

• For all x, y, and z, if x ≲ y and y ≲ z then x ≲ z (transitivity).

• For all x and y, x ≲ y or y ≲ x (totality).

• Hence, for all x, x ≲ x (reflexivity).

A total order is a total preorder which is antisymmetric, in other words, which is also a partial order. Total preordersare sometimes also called preference relations.The complement of a strict weak order is a total preorder, and vice versa, but it seems more natural to relate strictweak orders and total preorders in a way that preserves rather than reverses the order of the elements. Thus we takethe inverse of the complement: for a strict weak ordering <, define a total preorder ≲ by setting x ≲ y whenever it isnot the case that y < x. In the other direction, to define a strict weak ordering < from a total preorder ≲ , set x < ywhenever it is not the case that y ≲ x.[6]

In any preorder there is a corresponding equivalence relation where two elements x and y are defined as equivalent ifx ≲ y and y ≲ x. In the case of a total preorder the corresponding partial order on the set of equivalence classes is atotal order. Two elements are equivalent in a total preorder if and only if they are incomparable in the correspondingstrict weak ordering.

75.2.3 Ordered partitions

A partition of a set S is a family of disjoint subsets of S that have S as their union. A partition, together with atotal order on the sets of the partition, gives a structure called by Richard P. Stanley an ordered partition[7] and byTheodore Motzkin a list of sets.[8] An ordered partition of a finite set may be written as a finite sequence of the setsin the partition: for instance, the three ordered partitions of the set {a, b} are

{a}, {b},{b}, {a}, and{a, b}.

In a strict weak ordering, the equivalence classes of incomparability give a set partition, in which the sets inherit atotal ordering from their elements, giving rise to an ordered partition. In the other direction, any ordered partitiongives rise to a strict weak ordering in which two elements are incomparable when they belong to the same set in thepartition, and otherwise inherit the order of the sets that contain them.







https://en.wikipedia.org/wiki/Preorder#Constructions




https://en.wikipedia.org/wiki/Theodore_Motzkin

https://en.wikipedia.org/wiki/Finite_sequence

75.3. RELATED TYPES OF ORDERING 273

75.2.4 Representation by functions

For sets of sufficiently small cardinality, a third axiomatization is possible, based on real-valued functions. If X is anyset and f a real-valued function on X then f induces a strict weak order on X by setting a < b if and only if f(a) < f(b).The associated total preorder is given by setting a ≲ b if and only if f(a) ≤ f(b), and the associated equivalence bysetting a ∼ b if and only if f(a) = f(b).The relations do not change when f is replaced by g o f (composite function), where g is a strictly increasing real-valued function defined on at least the range of f. Thus e.g. a utility function defines a preference relation. In thiscontext, weak orderings are also known as preferential arrangements.[9]

If X is finite or countable, every weak order on X can be represented by a function in this way.[10] However, thereexist strict weak orders that have no corresponding real function. For example, there is no such function for thelexicographic order on Rn. Thus, while in most preference relation models the relation defines a utility function up toorder-preserving transformations, there is no such function for lexicographic preferences.More generally, if X is a set, and Y is a set with a strict weak ordering "<", and f a function from X to Y, then finduces a strict weak ordering on X by setting a < b if and only if f(a) < f(b). As before, the associated total preorderis given by setting a ≲ b if and only if f(a) ≲ f(b), and the associated equivalence by setting a ∼ b if and onlyif f(a) ∼ f(b). It is not assumed here that f is an injective function, so a class of two equivalent elements on Ymay induce a larger class of equivalent elements on X. Also, f is not assumed to be an surjective function, so a classof equivalent elements on Y may induce a smaller or empty class on X. However, the function f induces an injectivefunction that maps the partition on X to that on Y. Thus, in the case of finite partitions, the number of classes in X isless than or equal to the number of classes on Y.

75.3 Related types of ordering

Semiorders generalize strict weak orderings, but do not assume transitivity of incomparability.[11] A strict weak orderthat is trichotomous is called a strict total order.[12] The total preorder which is the inverse of its complement is inthis case a total order.For a strict weak order "<" another associated reflexive relation is its reflexive closure, a (non-strict) partial order "≤".The two associated reflexive relations differ with regard to different a and b for which neither a < b nor b < a: in thetotal preorder corresponding to a strict weak order we get a ≲ b and b ≲ a, while in the partial order given by thereflexive closure we get neither a ≤ b nor b ≤ a. For strict total orders these two associated reflexive relations arethe same: the corresponding (non-strict) total order.[12] The reflexive closure of a strict weak ordering is a type ofseries-parallel partial order.

75.4 All weak orders on a finite set

75.4.1 Combinatorial enumeration

Main article: ordered Bell number

The number of distinct weak orders (represented either as strict weak orders or as total preorders) on an n-elementset is given by the following sequence (sequence A000670 in OEIS):These numbers are also called the Fubini numbers or ordered Bell numbers.For example, for a set of three labeled items, there is one weak order in which all three items are tied. There are threeways of partitioning the items into one singleton set and one group of two tied items, and each of these partitionsgives two weak orders (one in which the singleton is smaller than the group of two, and one in which this ordering isreversed), giving six weak orders of this type. And there is a single way of partitioning the set into three singletons,which can be totally ordered in six different ways. Thus, altogether, there are 13 different weak orders on three items.



https://en.wikipedia.org/wiki/Monotonic_function

https://en.wikipedia.org/wiki/Utility#Utility_functions


https://en.wikipedia.org/wiki/Lexicographic_order


https://en.wikipedia.org/wiki/Lexicographic_preferences







https://en.wikipedia.org/wiki/Series-parallel_partial_order

https://en.wikipedia.org/wiki/Ordered_Bell_number




274 CHAPTER 75. WEAK ORDERING

(4,1,2,3)(4,2,1,3)

(3,2,1,4)

(3,1,2,4)

(2,1,3,4)

(1,2,3,4)

(1,2,4,3)

(1,3,2,4)

(2,1,4,3)

(2,3,1,4)

(3,1,4,2)

(4,1,3,2)

(4,2,3,1)

(3,2,4,1)(2,4,1,3)

(1,4,2,3)

(1,3,4,2)

(2,3,4,1)

(1,4,3,2)

(2,4,3,1)

(3,4,2,1)

(4,3,2,1)

(4,3,1,2)

(3,4,1,2)

The permutohedron on four elements, a three-dimensional convex polyhedron. It has 24 vertices, 36 edges, and 14 two-dimensionalfaces, which all together with the whole three-dimensional polyhedron correspond to the 75 weak orderings on four elements.

75.4.2 Adjacency structure

Unlike for partial orders, the family of weak orderings on a given finite set is not in general connected by moves thatadd or remove a single order relation to a given ordering. For instance, for three elements, the ordering in which allthree elements are tied differs by at least two pairs from any other weak ordering on the same set, in either the strictweak ordering or total preorder axiomatizations. However, a different kind of move is possible, in which the weakorderings on a set are more highly connected. Define a dichotomy to be a weak ordering with two equivalence classes,and define a dichotomy to be compatible with a given weak ordering if every two elements that are related in theordering are either related in the same way or tied in the dichotomy. Alternatively, a dichotomy may be defined as aDedekind cut for a weak ordering. Then a weak ordering may be characterized by its set of compatible dichotomies.For a finite set of labeled items, every pair of weak orderings may be connected to each other by a sequence of movesthat add or remove one dichotomy at a time to or from this set of dichotomies. Moreover, the undirected graph thathas the weak orderings as its vertices, and these moves as its edges, forms a partial cube.[13]

Geometrically, the total orderings of a given finite set may be represented as the vertices of a permutohedron, and thedichotomies on this same set as the facets of the permutohedron. In this geometric representation, the weak orderingson the set correspond to the faces of all different dimensions of the permutohedron (including the permutohedronitself, but not the empty set, as a face). The codimension of a face gives the number of equivalence classes in thecorresponding weak ordering.[14] In this geometric representation the partial cube of moves on weak orderings is thegraph describing the covering relation of the face lattice of the permutohedron.For instance, for n = 3, the permutohedron on three elements is just a regular hexagon. The face lattice of the hexagon(again, including the hexagon itself as a face, but not including the empty set) has thirteen elements: one hexagon,six edges, and six vertices, corresponding to the one completely tied weak ordering, six weak orderings with one tie,and six total orderings. The graph of moves on these 13 weak orderings is shown in the figure.

https://en.wikipedia.org/wiki/Convex_polyhedron

https://en.wikipedia.org/wiki/Dedekind_cut


https://en.wikipedia.org/wiki/Partial_cube

https://en.wikipedia.org/wiki/Permutohedron

https://en.wikipedia.org/wiki/Codimension


https://en.wikipedia.org/wiki/Face_lattice

75.5. APPLICATIONS 275

75.5 Applications

As mentioned above, weak orders have applications in utility theory.[10] In linear programming and other types ofcombinatorial optimization problem, the prioritization of solutions or of bases is often given by a weak order, de-termined by a real-valued objective function; the phenomenon of ties in these orderings is called “degeneracy”, andseveral types of tie-breaking rule have been used to refine this weak ordering into a total ordering in order to preventproblems caused by degeneracy.[15]

Weak orders have also been used in computer science, in partition refinement based algorithms for lexicographicbreadth-first search and lexicographic topological ordering. In these algorithms, a weak ordering on the vertices of agraph (represented as a family of sets that partition the vertices, together with a doubly linked list providing a totalorder on the sets) is gradually refined over the course of the algorithm, eventually producing a total ordering that isthe output of the algorithm.[16]

In the Standard Library for the C++ programming language, the set and multiset data types sort their input by acomparison function that is specified at the time of template instantiation, and that is assumed to implement a strictweak ordering.[17]

75.6 References[1] Roberts, Fred; Tesman, Barry (2011), Applied Combinatorics (2nd ed.), CRC Press, Section 4.2.4 Weak Orders, pp. 254–

256, ISBN 9781420099836.

[2] de Koninck, J. M. (2009), Those Fascinating Numbers, American Mathematical Society, p. 4, ISBN 9780821886311.

[3] Baker, Kent (April 29, 2007), “The Bruce hangs on for Hunt Cup victory: Bug River, Lear Charm finish in dead heat forsecond”, The Baltimore Sun, (subscription required (help)).

[4] Regenwetter, Michel (2006), Behavioral Social Choice: Probabilistic Models, Statistical Inference, and Applications, Cam-bridge University Press, pp. 113ff, ISBN 9780521536660.


[6] Ehrgott, Matthias (2005), Multicriteria Optimization, Springer, Proposition 1.9, p. 10, ISBN 9783540276593.

[7] Stanley, Richard P. (1997), Enumerative Combinatorics, Vol. 2, Cambridge Studies in Advanced Mathematics 62, Cam-bridge University Press, p. 297.

[8] Motzkin, Theodore S. (1971), “Sorting numbers for cylinders and other classification numbers”, Combinatorics (Proc.Sympos. Pure Math., Vol. XIX, Univ. California, Los Angeles, Calif., 1968), Providence, R.I.: Amer. Math. Soc., pp.167–176, MR 0332508.

[9] Gross, O. A. (1962), “Preferential arrangements”, The American Mathematical Monthly 69: 4–8, doi:10.2307/2312725,MR 0130837.

[10] Roberts, Fred S. (1979), Measurement Theory, with Applications to Decisionmaking, Utility, and the Social Sciences, Ency-clopedia of Mathematics and its Applications 7, Addison-Wesley, Theorem 3.1, ISBN 978-0-201-13506-0.

[11] Luce, R. Duncan (1956), “Semiorders and a theory of utility discrimination”, Econometrica 24: 178–191, JSTOR 1905751,MR 0078632.

[12] Velleman, Daniel J. (2006), How to Prove It: A Structured Approach, Cambridge University Press, p. 204, ISBN 9780521675994.

[13] Eppstein, David; Falmagne, Jean-Claude; Ovchinnikov, Sergei (2008), Media Theory: Interdisciplinary Applied Mathemat-ics, Springer, Section 9.4, Weak Orders and Cubical Complexes, pp. 188–196.

[14] Ziegler, Günter M. (1995), Lectures on Polytopes, Graduate Texts in Mathematics 152, Springer, p. 18.

[15] Chvátal, Vašek (1983), Linear Programming, Macmillan, pp. 29–38, ISBN 9780716715870.

[16] Habib, Michel; Paul, Christophe; Viennot, Laurent (1999), “Partition refinement techniques: an interesting algorithmictool kit”, International Journal of Foundations of Computer Science 10 (2): 147–170, doi:10.1142/S0129054199000125,MR 1759929.

[17] Josuttis, Nicolai M. (2012), The C++ Standard Library: A Tutorial and Reference, Addison-Wesley, p. 469, ISBN9780132977739.

https://en.wikipedia.org/wiki/Linear_programming

https://en.wikipedia.org/wiki/Combinatorial_optimization

https://en.wikipedia.org/wiki/Objective_function



https://en.wikipedia.org/wiki/Lexicographic_breadth-first_search

https://en.wikipedia.org/wiki/Lexicographic_breadth-first_search

https://en.wikipedia.org/wiki/Coffman%E2%80%93Graham_algorithm


https://en.wikipedia.org/wiki/Doubly_linked_list

https://en.wikipedia.org/wiki/C++_Standard_Library

https://en.wikipedia.org/wiki/C++

https://en.wikipedia.org/wiki/Associative_containers_(C++)



http://books.google.com/books?id=qYuC1WsDKq8C&pg=PA4



http://www.highbeam.com/doc/1G1-162753665.html

http://www.highbeam.com/doc/1G1-162753665.html

https://en.wikipedia.org/wiki/The_Baltimore_Sun



http://www.karlin.mff.cuni.cz/~jezek/120/transitive1.pdf?

http://books.google.com/books?id=AwRjo6iP4_UC&pg=PA10




https://en.wikipedia.org/wiki/Theodore_Motzkin




https://dx.doi.org/10.2307%252F2312725



https://en.wikipedia.org/wiki/Fred_S._Roberts



https://en.wikipedia.org/wiki/R._Duncan_Luce





http://books.google.com/books?id=lptwaMuAtBAC&pg=PA204



https://en.wikipedia.org/wiki/David_Eppstein

https://en.wikipedia.org/wiki/Jean-Claude_Falmagne

https://en.wikipedia.org/wiki/G%C3%BCnter_M._Ziegler

https://en.wikipedia.org/wiki/Va%C5%A1ek_Chv%C3%A1tal

http://books.google.com/books?id=DN20_tW_BV0C&pg=PA29







http://books.google.com/books?id=9DEJKhasp7gC&pg=PT469



Chapter 76

Well-founded relation

“Noetherian induction” redirects here. For the use in topology, see Noetherian topological space.

In mathematics, a binary relation, R, is well-founded (or wellfounded) on a class X if and only if every non-emptysubset S X has a minimal element; that is, some element m of any S is not related by sRm (for instance, "m is notsmaller than”) for the rest of the s ∈ S.

∀S ⊆ X (S ̸= ∅ → ∃m ∈ S ∀s ∈ S (s,m) /∈ R)

(Some authors include an extra condition that R is set-like, i.e., that the elements less than any given element form aset.)Equivalently, assuming some choice, a relation is well-founded if and only if it contains no countable infinite descend-ing chains: that is, there is no infinite sequence x0, x1, x2, ... of elements of X such that xn₊₁ R x for every naturalnumber n.In order theory, a partial order is called well-founded if the corresponding strict order is a well-founded relation. Ifthe order is a total order then it is called a well-order.In set theory, a set x is called a well-founded set if the set membership relation is well-founded on the transitiveclosure of x. The axiom of regularity, which is one of the axioms of Zermelo–Fraenkel set theory, asserts that all setsare well-founded.A relation R is converse well-founded, upwards well-founded or Noetherian on X, if the converse relation R−1

is well-founded on X. In this case R is also said to satisfy the ascending chain condition. In the context of rewritingsystems, a Noetherian relation is also called terminating.

76.1 Induction and recursion

An important reason that well-founded relations are interesting is because a version of transfinite induction can beused on them: if (X, R) is a well-founded relation, P(x) is some property of elements of X, and we want to show that

P(x) holds for all elements x of X,

it suffices to show that:

If x is an element of X and P(y) is true for all y such that y R x, then P(x) must also be true.

That is,

∀x ∈ X [(∀y ∈ X (y Rx→ P (y))) → P (x)] → ∀x ∈ X P (x).

276

https://en.wikipedia.org/wiki/Noetherian_topological_space






https://en.wikipedia.org/wiki/Minimal_element




https://en.wikipedia.org/wiki/Infinite_descending_chain

https://en.wikipedia.org/wiki/Infinite_descending_chain








https://en.wikipedia.org/wiki/Transitive_set

https://en.wikipedia.org/wiki/Transitive_set

https://en.wikipedia.org/wiki/Axiom_of_regularity


https://en.wikipedia.org/wiki/Converse_relation

https://en.wikipedia.org/wiki/Ascending_chain_condition

https://en.wikipedia.org/wiki/Rewriting

https://en.wikipedia.org/wiki/Transfinite_induction

76.2. EXAMPLES 277

Well-founded induction is sometimes called Noetherian induction,[1] after Emmy Noether.On par with induction, well-founded relations also support construction of objects by transfinite recursion. Let (X,R) be a set-like well-founded relation and F a function that assigns an object F(x, g) to each pair of an element x ∈ Xand a function g on the initial segment {y: y R x} of X. Then there is a unique function G such that for every x ∈ X,

G(x) = F (x,G|{y:y Rx})

That is, if we want to construct a function G on X, we may define G(x) using the values of G(y) for y R x.As an example, consider the well-founded relation (N, S), where N is the set of all natural numbers, and S is the graphof the successor function x → x + 1. Then induction on S is the usual mathematical induction, and recursion on Sgives primitive recursion. If we consider the order relation (N, <), we obtain complete induction, and course-of-valuesrecursion. The statement that (N, <) is well-founded is also known as the well-ordering principle.There are other interesting special cases of well-founded induction. When the well-founded relation is the usualordering on the class of all ordinal numbers, the technique is called transfinite induction. When the well-founded setis a set of recursively-defined data structures, the technique is called structural induction. When the well-foundedrelation is set membership on the universal class, the technique is known as ∈-induction. See those articles for moredetails.

76.2 Examples

Well-founded relations which are not totally ordered include:

• the positive integers {1, 2, 3, ...}, with the order defined by a < b if and only if a divides b and a ≠ b.

• the set of all finite strings over a fixed alphabet, with the order defined by s < t if and only if s is a propersubstring of t.

• the set N × N of pairs of natural numbers, ordered by (n1, n2) < (m1, m2) if and only if n1 < m1 and n2 < m2.

• the set of all regular expressions over a fixed alphabet, with the order defined by s < t if and only if s is a propersubexpression of t.

• any class whose elements are sets, with the relation ∈ (“is an element of”). This is the axiom of regularity.

• the nodes of any finite directed acyclic graph, with the relation R defined such that a R b if and only if there isan edge from a to b.

Examples of relations that are not well-founded include:

• the negative integers {−1, −2, −3, …}, with the usual order, since any unbounded subset has no least element.

• The set of strings over a finite alphabet with more than one element, under the usual (lexicographic) order,since the sequence “B” > “AB” > “AAB” > “AAAB” > … is an infinite descending chain. This relation failsto be well-founded even though the entire set has a minimum element, namely the empty string.

• the rational numbers (or reals) under the standard ordering, since, for example, the set of positive rationals (orreals) lacks a minimum.

76.3 Other properties

If (X, <) is a well-founded relation and x is an element of X, then the descending chains starting at x are all finite, butthis does not mean that their lengths are necessarily bounded. Consider the following example: Let X be the union ofthe positive integers and a new element ω, which is bigger than any integer. Then X is a well-founded set, but thereare descending chains starting at ω of arbitrary great (finite) length; the chain ω, n − 1, n − 2, ..., 2, 1 has length nfor any n.

https://en.wikipedia.org/wiki/Emmy_Noether

https://en.wikipedia.org/wiki/Transfinite_recursion


https://en.wikipedia.org/wiki/Initial_segment


https://en.wikipedia.org/wiki/Mathematical_induction

https://en.wikipedia.org/wiki/Primitive_recursive_functions

https://en.wikipedia.org/wiki/Complete_induction

https://en.wikipedia.org/wiki/Course-of-values_recursion

https://en.wikipedia.org/wiki/Course-of-values_recursion

https://en.wikipedia.org/wiki/Well-ordering_principle

https://en.wikipedia.org/wiki/Ordinal_numbers


https://en.wikipedia.org/wiki/Structural_induction

https://en.wikipedia.org/wiki/%E2%88%88-induction




https://en.wikipedia.org/wiki/String_(computer_science)



https://en.wikipedia.org/wiki/Regular_expression

https://en.wikipedia.org/wiki/Axiom_of_regularity


https://en.wikipedia.org/wiki/Lexicographic_ordering


https://en.wikipedia.org/wiki/Real_numbers

278 CHAPTER 76. WELL-FOUNDED RELATION

The Mostowski collapse lemma implies that set membership is a universal among the extensional well-founded rela-tions: for any set-like well-founded relation R on a class X which is extensional, there exists a class C such that (X,R)is isomorphic to (C,∈).

76.4 Reflexivity

A relation R is said to be reflexive if a R a holds for every a in the domain of the relation. Every reflexive relation on anonempty domain has infinite descending chains, because any constant sequence is a descending chain. For example,in the natural numbers with their usual order ≤, we have 1 ≥ 1 ≥ 1 ≥ · · · . To avoid these trivial descendingsequences, when working with a reflexive relation R it is common to use (perhaps implicitly) the alternate relation R′defined such that a R′ b if and only if a R b and a ≠ b. In the context of the natural numbers, this means that therelation <, which is well-founded, is used instead of the relation ≤, which is not. In some texts, the definition of awell-founded relation is changed from the definition above to include this convention.

76.5 References[1] Bourbaki, N. (1972) Elements of mathematics. Commutative algebra, Addison-Wesley.

• Just, Winfried and Weese, Martin, Discovering Modern Set theory. I, American Mathematical Society (1998)ISBN 0-8218-0266-6.

https://en.wikipedia.org/wiki/Mostowski_collapse



Chapter 77

Well-order

In mathematics, a well-order relation (or well-ordering) on a set S is a total order on S with the property that everynon-empty subset of S has a least element in this ordering. The set S together with the well-order relation is thencalled awell-ordered set. The hyphen is frequently omitted in contemporary papers, yielding the spellingswellorder,wellordered, and wellordering.Every non-empty well-ordered set has a least element. Every element s of a well-ordered set, except a possible greatestelement, has a unique successor (next element), namely the least element of the subset of all elements greater thans. There may be elements besides the least element which have no predecessor (see Natural numbers below for anexample). In a well-ordered set S, every subset T which has an upper bound has a least upper bound, namely the leastelement of the subset of all upper bounds of T in S.If ≤ is a non-strict well-ordering, then < is a strict well-ordering. A relation is a strict well-ordering if and only if it isa well-founded strict total order. The distinction between strict and non-strict well-orders is often ignored since theyare easily interconvertible.Every well-ordered set is uniquely order isomorphic to a unique ordinal number, called the order type of the well-ordered set. The well-ordering theorem, which is equivalent to the axiom of choice, states that every set can bewell-ordered. If a set is well-ordered (or even if it merely admits a wellfounded relation), the proof technique oftransfinite induction can be used to prove that a given statement is true for all elements of the set.The observation that the natural numbers are well-ordered by the usual less-than relation is commonly called thewell-ordering principle (for natural numbers).

77.1 Ordinal numbers

Main article: Ordinal number

Every well-ordered set is uniquely order isomorphic to a unique ordinal number, called the order type of the well-ordered set. The position of each element within the ordered set is also given by an ordinal number. In the case of afinite set, the basic operation of counting, to find the ordinal number of a particular object, or to find the object witha particular ordinal number, corresponds to assigning ordinal numbers one by one to the objects. The size (numberof elements, cardinal number) of a finite set is equal to the order type. Counting in the everyday sense typically startsfrom one, so it assigns to each object the size of the initial segment with that object as last element. Note that thesenumbers are one more than the formal ordinal numbers according to the isomorphic order, because these are equalto the number of earlier objects (which corresponds to counting from zero). Thus for finite n, the expression "n-thelement” of a well-ordered set requires context to know whether this counts from zero or one. In a notation "β-thelement” where β can also be an infinite ordinal, it will typically count from zero.For an infinite set the order type determines the cardinality, but not conversely: well-ordered sets of a particularcardinality can have many different order types. For a countably infinite set, the set of possible order types is evenuncountable.

279




https://en.wikipedia.org/wiki/Non-empty





https://en.wikipedia.org/wiki/Well-order#Natural_numbers


https://en.wikipedia.org/wiki/Non-strict_order


https://en.wikipedia.org/wiki/Strict_total_order




https://en.wikipedia.org/wiki/Well-ordering_theorem


https://en.wikipedia.org/wiki/Wellfounded_relation



https://en.wikipedia.org/wiki/Well-ordering_principle





https://en.wikipedia.org/wiki/Counting



280 CHAPTER 77. WELL-ORDER

77.2 Examples and counterexamples

77.2.1 Natural numbers

The standard ordering ≤ of the natural numbers is a well-ordering and has the additional property that every non-zeronatural number has a unique predecessor.Another well-ordering of the natural numbers is given by defining that all even numbers are less than all odd numbers,and the usual ordering applies within the evens and the odds:

0 2 4 6 8 ... 1 3 5 7 9 ...

This is a well-ordered set of order type ω + ω. Every element has a successor (there is no largest element). Twoelements lack a predecessor: 0 and 1.

77.2.2 Integers

Unlike the standard ordering ≤ of the natural numbers, the standard ordering ≤ of the integers is not a well-ordering,since, for example, the set of negative integers does not contain a least element.The following relation R is an example of well-ordering of the integers: x R y if and only if one of the followingconditions holds:

1. x = 0

2. x is positive, and y is negative

3. x and y are both positive, and x ≤ y

4. x and y are both negative, and |x| ≤ |y|

This relation R can be visualized as follows:

0 1 2 3 4 ... −1 −2 −3 ...

R is isomorphic to the ordinal number ω + ω.Another relation for well-ordering the integers is the following definition: x ≤ y iff (|x| < |y| or (|x| = |y| and x ≤ y)).This well-order can be visualized as follows:

0 −1 1 −2 2 −3 3 −4 4 ...

This has the order type ω.

77.2.3 Reals

The standard ordering ≤ of the positive real numbers is not a well-ordering, since, for example, the open interval (0,1) does not contain a least element. From the ZFC axioms of set theory (including the axiom of choice) one can showthat there is a well-order of the reals. Also Wacław Sierpiński proved that ZF + GCH (the generalized continuumhypothesis) imply the axiom of choice and hence a well-order of the reals. Nonetheless, it is possible to show thatthe ZFC+GCH axioms alone are not sufficient to prove the existence of a definable (by a formula) well-order ofthe reals.[1] However it is consistent with ZFC that a definable well-ordering of the reals exists—for example, it isconsistent with ZFC that V=L, and it follows from ZFC+V=L that a particular formula well-orders the reals, or indeedany set.An uncountable subset of the real numbers with the standard ordering ≤ cannot be a well-order: Suppose X is a subsetof R well-ordered by ≤. For each x in X, let s(x) be the successor of x in ≤ ordering on X (unless x is the last elementof X). Let A = { (x, s(x)) | x ∈ X } whose elements are nonempty and disjoint intervals. Each such interval contains




https://en.wikipedia.org/wiki/Negative_number







https://en.wikipedia.org/wiki/Open_interval



https://en.wikipedia.org/wiki/Wac%C5%82aw_Sierpi%C5%84ski

https://en.wikipedia.org/wiki/Generalized_continuum_hypothesis

https://en.wikipedia.org/wiki/Generalized_continuum_hypothesis

https://en.wikipedia.org/wiki/V=L

77.3. EQUIVALENT FORMULATIONS 281

at least one rational number, so there is an injective function from A to Q. There is an injection from X to A (exceptpossibly for a last element of X which could be mapped to zero later). And it is well known that there is an injectionfrom Q to the natural numbers (which could be chosen to avoid hitting zero). Thus there is an injection from X to thenatural numbers which means that X is countable. On the other hand, a countably infinite subset of the reals may ormay not be a well-order with the standard "≤".

• The natural numbers are a well-order.• The set {1/n : n =1,2,3,...} has no least element and is therefore not a well-order.

Examples of well-orders:

• The set of numbers { − 2−n | 0 ≤ n < ω } has order type ω.• The set of numbers { − 2−n − 2−m−n | 0 ≤ m,n < ω } has order type ω². The previous set is the set of limit

points within the set. Within the set of real numbers, either with the ordinary topology or the order topology,0 is also a limit point of the set. It is also a limit point of the set of limit points.

• The set of numbers { − 2−n | 0 ≤ n < ω } ∪ { 1 } has order type ω + 1. With the order topology of this set, 1 isa limit point of the set. With the ordinary topology (or equivalently, the order topology) of the real numbers itis not.

77.3 Equivalent formulations

If a set is totally ordered, then the following are equivalent to each other:

1. The set is well-ordered. That is, every nonempty subset has a least element.2. Transfinite induction works for the entire ordered set.3. Every strictly decreasing sequence of elements of the set must terminate after only finitely many steps (assuming

the axiom of dependent choice).4. Every subordering is isomorphic to an initial segment.

77.4 Order topology

Every well-ordered set can be made into a topological space by endowing it with the order topology.With respect to this topology there can be two kinds of elements:

• isolated points - these are the minimum and the elements with a predecessor.• limit points - this type does not occur in finite sets, and may or may not occur in an infinite set; the infinite sets

without limit point are the sets of order type ω, for example N.

For subsets we can distinguish:

• Subsets with a maximum (that is, subsets which are bounded by themselves); this can be an isolated point or alimit point of the whole set; in the latter case it may or may not be also a limit point of the subset.

• Subsets which are unbounded by themselves but bounded in the whole set; they have no maximum, but asupremum outside the subset; if the subset is non-empty this supremum is a limit point of the subset and hencealso of the whole set; if the subset is empty this supremum is the minimum of the whole set.

• Subsets which are unbounded in the whole set.

A subset is cofinal in the whole set if and only if it is unbounded in the whole set or it has a maximum which is alsomaximum of the whole set.A well-ordered set as topological space is a first-countable space if and only if it has order type less than or equal toω1 (omega-one), that is, if and only if the set is countable or has the smallest uncountable order type.


https://en.wikipedia.org/wiki/Limit_point





https://en.wikipedia.org/wiki/Axiom_of_dependent_choice



https://en.wikipedia.org/wiki/Isolated_point


https://en.wikipedia.org/wiki/Bounded_set#Boundedness_in_order_theory

https://en.wikipedia.org/wiki/Cofinal_(mathematics)

https://en.wikipedia.org/wiki/First-countable_space

https://en.wikipedia.org/wiki/First_uncountable_ordinal

https://en.wikipedia.org/wiki/Countable

https://en.wikipedia.org/wiki/Uncountable

282 CHAPTER 77. WELL-ORDER

77.5 See also• Tree (set theory), generalization

• Well-ordering theorem

• Ordinal number

• Well-founded set

• Well partial order

• Prewellordering

• Directed set

77.6 References[1] S. Feferman: “Some Applications of the Notions of Forcing and Generic Sets”, Fundamenta Mathematicae, 56 (1964)

325-345

• Folland, Gerald B. (1999). Real Analysis: Modern Techniques and Their Applications. Pure and applied math-ematics (2nd ed.). John Wiley & Sons. pp. 4–6, 9. ISBN 978-0-471-31716-6.

https://en.wikipedia.org/wiki/Tree_(set_theory)

https://en.wikipedia.org/wiki/Well-ordering_theorem


https://en.wikipedia.org/wiki/Well-founded_set

https://en.wikipedia.org/wiki/Well_partial_order



https://en.wikipedia.org/wiki/Solomon_Feferman

https://en.wikipedia.org/wiki/Gerald_Folland

https://en.wikipedia.org/wiki/John_Wiley_&_Sons



Chapter 78

Well-quasi-ordering

In mathematics, specifically order theory, a well-quasi-ordering or wqo is a quasi-ordering that is well-founded,meaning that any infinite sequence of elements x0 , x1 , x2 , … from X contains an increasing pair xi ≤ xj withi < j .

78.1 Motivation

Well-founded induction can be used on any set with a well-founded relation, thus one is interested in when a quasi-order is well-founded. However the class of well-founded quasiorders is not closed under certain operations - thatis, when a quasi-order is used to obtain a new quasi-order on a set of structures derived from our original set, thisquasiorder is found to be not well-founded. By placing stronger restrictions on the original well-founded quasiorderingone can hope to ensure that our derived quasiorderings are still well-founded.An example of this is the power set operation. Given a quasiordering ≤ for a set X one can define a quasiorder ≤+

on X 's power set P (X) by setting A ≤+ B if and only if for each element of A one can find some element of Bwhich is larger than it under ≤ . One can show that this quasiordering on P (X) needn't be well-founded, but if onetakes the original quasi-ordering to be a well-quasi-ordering, then it is.


A well-quasi-ordering on a setX is a quasi-ordering (i.e., a reflexive, transitive binary relation) such that any infinitesequence of elements x0 , x1 , x2 , … from X contains an increasing pair xi ≤ xj with i < j . The set X is said tobe well-quasi-ordered, or shortly wqo.A well partial order, or a wpo, is a wqo that is a proper ordering relation, i.e., it is antisymmetric.Among other ways of defining wqo’s, one is to say that they are quasi-orderings which do not contain infinite strictlydecreasing sequences (of the form x0 > x1 > x2 >…) nor infinite sequences of pairwise incomparable elements.Hence a quasi-order ( X ,≤) is wqo if and only if it is well-founded and has no infinite antichains.

78.3 Examples

• (N,≤) , the set of natural numbers with standard ordering, is a well partial order (in fact, a well-order). How-ever, (Z,≤) , the set of positive and negative integers, is not a well-quasi-order, because it is not well-founded.

• (N, |) , the set of natural numbers ordered by divisibility, is not a well partial order: the prime numbers are aninfinite antichain.

• (Nk,≤) , the set of vectors of k natural numbers (where k is finite) with component-wise ordering, is a wellpartial order (Dickson’s lemma). More generally, if (X,≤) is well-quasi-order, then (Xk,≤k) is also a well-quasi-order for all k .

283




https://en.wikipedia.org/wiki/Well-founded

https://en.wikipedia.org/wiki/Infinity

https://en.wikipedia.org/wiki/Well-founded_induction






https://en.wikipedia.org/wiki/Infinity






https://en.wikipedia.org/wiki/Dickson%2527s_lemma

284 CHAPTER 78. WELL-QUASI-ORDERING

• LetX be an arbitrary finite set with at least two elements. The setX∗ of words overX ordered lexicographically(as in a dictionary) isnot a well-quasi-order because it contains the infinite decreasing sequence b, ab, aab, aaab, . . .. Similarly,X∗ ordered by the prefix relation is not a well-quasi-order, because the previous sequence is an in-finite antichain of this partial order. However,X∗ ordered by the subsequence relation is a well partial order.[1]

(If X has only one element, these three partial orders are identical.)

• More generally, (X∗,≤) , the set of finite X -sequences ordered by embedding is a well-quasi-order if andonly if (X,≤) is a well-quasi-order (Higman’s lemma). Recall that one embeds a sequence u into a sequence vby finding a subsequence of v that has the same length as u and that dominates it term by term. When (X,=)is a finite unordered set, u ≤ v if and only if u is a subsequence of v .

• (Xω,≤) , the set of infinite sequences over a well-quasi-order (X,≤) , ordered by embedding, is not a well-quasi-order in general. That is, Higman’s lemma does not carry over to infinite sequences. Better-quasi-orderings have been introduced to generalize Higman’s lemma to sequences of arbitrary lengths.

• Embedding between finite trees with nodes labeled by elements of a wqo (X,≤) is a wqo (Kruskal’s treetheorem).

• Embedding between infinite trees with nodes labeled by elements of a wqo (X,≤) is a wqo (Nash-Williams'theorem).

• Embedding between countable scattered linear order types is a well-quasi-order (Laver's theorem).

• Embedding between countable boolean algebras is a well-quasi-order. This follows from Laver’s theorem anda theorem of Ketonen.

• Finite graphs ordered by a notion of embedding called "graph minor" is a well-quasi-order (Robertson–Seymourtheorem).

• Graphs of finite tree-depth ordered by the induced subgraph relation form a well-quasi-order,[2] as do thecographs ordered by induced subgraphs.[3]

78.4 Wqo’s versus well partial orders

In practice, the wqo’s one manipulates are quite often not orderings (see examples above), and the theory is technicallysmoother if we do not require antisymmetry, so it is built with wqo’s as the basic notion.Observe that a wpo is a wqo, and that a wqo gives rise to a wpo between equivalence classes induced by the kernelof the wqo. For example, if we order Z by divisibility, we end up with n ≡ m if and only if n = ±m , so that(Z, |) ≈ (N, |) .

78.5 Infinite increasing subsequences

If ( X , ≤) is wqo then every infinite sequence x0 , x1 , x2 , … contains an infinite increasing subsequence xn0 ≤xn1 ≤ xn2 ≤… (with n0 < n1 < n2 <…). Such a subsequence is sometimes called perfect. This can be proved bya Ramsey argument: given some sequence (xi)i , consider the set I of indexes i such that xi has no larger or equalxj to its right, i.e., with i < j . If I is infinite, then the I -extracted subsequence contradicts the assumption that Xis wqo. So I is finite, and any xn with n larger than any index in I can be used as the starting point of an infiniteincreasing subsequence.The existence of such infinite increasing subsequences is sometimes taken as a definition for well-quasi-ordering,leading to an equivalent notion.

78.6 Properties of wqos

• Given a quasiordering (X,≤) the quasiordering (P (X),≤+) defined byA ≤+ B ⇐⇒ ∀a ∈ A∃b ∈ B(a ≤b) is well-founded if and only if (X,≤) is a wqo.[4]

https://en.wikipedia.org/wiki/Formal_language#Words_over_an_alphabet


https://en.wikipedia.org/wiki/Prefix_(computer_science)#Prefix

https://en.wikipedia.org/wiki/Subsequence


https://en.wikipedia.org/wiki/Higman%2527s_lemma

https://en.wikipedia.org/wiki/Better-quasi-ordering


https://en.wikipedia.org/wiki/Kruskal%2527s_tree_theorem

https://en.wikipedia.org/wiki/Kruskal%2527s_tree_theorem

https://en.wikipedia.org/wiki/Crispin_St._J._A._Nash-Williams

https://en.wikipedia.org/wiki/Scattered_order

https://en.wikipedia.org/wiki/Linear_order

https://en.wikipedia.org/wiki/Richard_Laver

https://en.wikipedia.org/wiki/Boolean_algebras

https://en.wikipedia.org/wiki/Graph_minor

https://en.wikipedia.org/wiki/Robertson%E2%80%93Seymour_theorem

https://en.wikipedia.org/wiki/Robertson%E2%80%93Seymour_theorem

https://en.wikipedia.org/wiki/Tree-depth

https://en.wikipedia.org/wiki/Induced_subgraph


https://en.wikipedia.org/wiki/Ramsey_theory

78.7. NOTES 285

• A quasiordering is a wqo if and only if the corresponding partial order (obtained by quotienting by x ∼ y ⇐⇒x ≤ y ∧ y ≤ x ) has no infinite descending sequences or antichains. (This can be proved using a Ramseyargument as above.)

• Given a well-quasi-ordering (X,≤) , any sequence of subsets S0 ⊆ S1 ⊆ ... ⊆ X such that ∀i ∈ N, ∀x, y ∈X,x ≤ y∧x ∈ Si ⇒ y ∈ Si eventually stabilises (meaning there is an indexn ∈ N such thatSn = Sn+1 = ...; subsets S ⊆ X with the property ∀x, y ∈ X,x ≤ y ∧ x ∈ S ⇒ y ∈ S are usually called upward-closed):assuming the contrary ∀i ∈ N∃j ∈ N, j > i, ∃x ∈ Sj \Si , a contradiction is reached by extracting an infinitenon-ascending subsequence.

• Given a well-quasi-ordering (X,≤) , any subset S ⊆ X which is upward-closed with respect to ≤ has a finitenumber of minimal elements w.r.t. ≤ , for otherwise the minimal elements of S would constitute an infiniteantichain.

78.7 Notes[1] Gasarch, W. (1998), “A survey of recursive combinatorics”, Handbook of Recursive Mathematics, Vol. 2, Stud. Logic

Found. Math. 139, Amsterdam: North-Holland, pp. 1041–1176, doi:10.1016/S0049-237X(98)80049-9, MR 1673598.See in particular page 1160.

[2] Nešetřil, Jaroslav; Ossona de Mendez, Patrice (2012), “Lemma 6.13”, Sparsity: Graphs, Structures, and Algorithms, Algo-rithms and Combinatorics 28, Heidelberg: Springer, p. 137, doi:10.1007/978-3-642-27875-4, ISBN 978-3-642-27874-7,MR 2920058.

[3] Damaschke, Peter (1990), “Induced subgraphs and well-quasi-ordering”, Journal of Graph Theory 14 (4): 427–435,doi:10.1002/jgt.3190140406, MR 1067237.

[4] Forster, Thomas (2003). “Better-quasi-orderings and coinduction”. Theoretical Computer Science 309 (1–3): 111–123.doi:10.1016/S0304-3975(03)00131-2.

78.8 References• Dickson, L. E. (1913). “Finiteness of the odd perfect and primitive abundant numbers with r distinct prime

factors”. American Journal of Mathematics 35 (4): 413–422. doi:10.2307/2370405. JSTOR 2370405.

• Higman, G. (1952). “Ordering by divisibility in abstract algebras”. Proceedings of the London MathematicalSociety 2: 326–336. doi:10.1112/plms/s3-2.1.326.

• Kruskal, J. B. (1972). “The theory of well-quasi-ordering: A frequently discovered concept”. Journal ofCombinatorial Theory. Series A 13 (3): 297–305. doi:10.1016/0097-3165(72)90063-5.

• Ketonen, Jussi (1978). “The structure of countable Boolean algebras”. Annals of Mathematics 108 (1): 41–89.doi:10.2307/1970929. JSTOR 1970929.

• Milner, E. C. (1985). “Basic WQO- and BQO-theory”. In Rival, I. Graphs and Order. The Role of Graphs inthe Theory of Ordered Sets and Its Applications. D. Reidel Publishing Co. pp. 487–502. ISBN 90-277-1943-8.

• Gallier, Jean H. (1991). “What’s so special about Kruskal’s theorem and the ordinal Γo? A survey of some re-sults in proof theory”. Annals of Pure and Applied Logic 53 (3): 199–260. doi:10.1016/0168-0072(91)90022-E.

78.9 See also• Better-quasi-ordering

• Prewellordering

• Well-order




https://en.wikipedia.org/wiki/Closure_(mathematics)

https://en.wikipedia.org/wiki/William_Gasarch





https://en.wikipedia.org/wiki/Jaroslav_Ne%C5%A1et%C5%99il


https://dx.doi.org/10.1007%252F978-3-642-27875-4






https://dx.doi.org/10.1002%252Fjgt.3190140406



https://en.wikipedia.org/wiki/Thomas_Forster



https://en.wikipedia.org/wiki/Leonard_Dickson

https://en.wikipedia.org/wiki/American_Journal_of_Mathematics


https://dx.doi.org/10.2307%252F2370405



https://en.wikipedia.org/wiki/Graham_Higman


https://dx.doi.org/10.1112%252Fplms%252Fs3-2.1.326

https://en.wikipedia.org/wiki/Joseph_Kruskal




https://dx.doi.org/10.1016%252F0097-3165%252872%252990063-5

https://en.wikipedia.org/wiki/Annals_of_Mathematics


https://dx.doi.org/10.2307%252F1970929



https://en.wikipedia.org/wiki/Ivan_Rival




https://dx.doi.org/10.1016%252F0168-0072%252891%252990022-E

https://dx.doi.org/10.1016%252F0168-0072%252891%252990022-E





78.10 Text and image sources, contributors, and licenses

78.10.1 Text• Accessibility relation Source: https://en.wikipedia.org/wiki/Accessibility_relation?oldid=664002580Contributors: Michael Hardy, Wma-

han, Roisterer, Urhixidur, Spug, Firsfron, Nivaca, Bgwhite, KSchutte, Imz, Mhss, Lacatosias, Mets501, Simeon, Gregbard, Egriffin,Classicalecon, Auntof6, Alexbot, Sun Creator, Dthomsen8, Addbot, Frehley, Tempodivalse, MikeyMouse10, ISpamThisSite3, Widr,Cabbagist and Anonymous: 28

• Allegory (category theory) Source: https://en.wikipedia.org/wiki/Allegory_(category_theory)?oldid=636236531Contributors: The Anome,Crisófilax, Aempirei, Woohookitty, Mindmatrix, Jfr26, Sam Staton, Yobot, ComputScientist, RjwilmsiBot and Anonymous: 5

• Alternatization Source: https://en.wikipedia.org/wiki/Alternatization?oldid=674572806Contributors: Rich Farmbrough, Tompw, Vipul,BD2412, Malcolma, Elonka, Mclay1, David Eppstein, Haseldon, LokiClock, JP.Martin-Flatin, Erik9bot, Paine Ellsworth, Quondum,Solomon7968 and Anonymous: 1

• Ancestral relation Source: https://en.wikipedia.org/wiki/Ancestral_relation?oldid=668533172 Contributors: Zundark, Meloman, CBM,Palnot, AnomieBOT, Malts y, TricksterWolf, Jochen Burghardt and Anonymous: 5

• Antisymmetric relation Source: https://en.wikipedia.org/wiki/Antisymmetric_relation?oldid=653233446Contributors: AxelBoldt, Patrick,Michael Hardy, Charles Matthews, Dcoetzee, MathMartin, Rholton, Tobias Bergemann, Giftlite, Sam Hocevar, Lumidek, Nparikh, As-cánder, Paul August, El C, Spoon!, Rpresser, Jumbuck, EvenT, Adrian.benko, Isnow, Nivaca, Fresheneesz, Chobot, Roboto de Ajvol,Kelovy, Arthur Rubin, Wasseralm, SmackBot, RDBury, Incnis Mrsi, InverseHypercube, Jcarroll, Bluebot, Jdthood, Lambiam, DabMa-chine, Mike Fikes, Gregbard, WillowW, JAnDbot, TAnthony, Catskineater, Mark lee stillwell, Semmelweiss, PaulTanenbaum, Jackfork,Henry Delforn (old), DuaneLAnderson, P30Carl, Hakuku, Addbot, LaaknorBot, CarsracBot, Verbal, Legobot, Luckas-bot, Yobot, Xqbot,Adavis444, Erik9bot, De bezige bij, EmausBot, Theophil789, Joel B. Lewis, AvocatoBot, MadGuy7023, Alexjbest, Nbrader, Boga159and Anonymous: 19

• Asymmetric relation Source: https://en.wikipedia.org/wiki/Asymmetric_relation?oldid=653233105Contributors: Patrick, Charles Matthews,Dcoetzee, Brianjd, Paul August, Oliphaunt, Palica, Fresheneesz, Mditto, Arthur Rubin, SmackBot, Jdthood, Tsca.bot, Jklin, Lakinekaki,CRGreathouse, Gregbard, Alastair Haines, Magioladitis, Robin S, Jy00912345, Taifunbrowser, VolkovBot, Henry Delforn (old), Ben-der2k14, Addbot, Yobot, Erik9bot, WikitanvirBot, Theophil789, TheodoreYou, MerlIwBot, Helpful Pixie Bot, Lerutit and Anonymous:12

• Better-quasi-ordering Source: https://en.wikipedia.org/wiki/Better-quasi-ordering?oldid=640379072Contributors: Xezbeth, Chris Capoc-cia, Citation bot, FrescoBot, Gongfarmerzed and BG19bot

• Bidirectional transformation Source: https://en.wikipedia.org/wiki/Bidirectional_transformation?oldid=646599869Contributors: MichaelHardy, Bearcat, Raphink, Ruud Koot, Educres, Yobot, Winterst, Borkificator, Mr Sheep Measham, BG19bot and Tmslnz

• Bijection Source: https://en.wikipedia.org/wiki/Bijection?oldid=670090850 Contributors: Damian Yerrick, AxelBoldt, Tarquin, Jan Hid-ders, XJaM, Toby Bartels, Michael Hardy, Wshun, TakuyaMurata, GTBacchus, Karada, Александър, Glenn, Poor Yorick, Rob Hooft,Pizza Puzzle, Hashar, Hawthorn, Charles Matthews, Dcoetzee, Dysprosia, Hyacinth, David Shay, Ed g2s, Bevo, Robbot, Fredrik, Ben-wing, Bkell, Salty-horse, Tobias Bergemann, Giftlite, Jorge Stolfi, Alberto da Calvairate~enwiki, MarkSweep, Tsemii, Vivacissama-mente, Guanabot, Guanabot2, Quistnix, Paul August, Ignignot, MisterSheik, Nickj, Kevin Lamoreau, Obradovic Goran, Pearle, Hashar-Bot~enwiki, Dallashan~enwiki, ABCD, Schapel, Palica, MarSch, Salix alba, FlaBot, VKokielov, RexNL, Chobot, YurikBot, MichaelSlone, Member, SmackBot, RDBury, Mmernex, Octahedron80, Mhym, Bwoodacre, Dreadstar, Davipo, Loadmaster, Mets501, Drefty-mac, Hilverd, Johnfuhrmann, Bill Malloy, Domitori, JRSpriggs, CmdrObot, Gregbard, Yaris678, Sam Staton, Panzer raccoon!, Kilva,AbcXyz, Escarbot, Salgueiro~enwiki, JAnDbot, David Eppstein, Martynas Patasius, Paulnwatts, Cpiral, GaborLajos, Policron, Diegovb,UnicornTapestry, Yomcat, Wykypydya, Bongoman666, SieBot, Paradoctor, Paolo.dL, Smaug123, MiNombreDeGuerra, JackSchmidt,I Spel Good~enwiki, Peiresc~enwiki, Classicalecon, Adrianwn, Biagioli, Watchduck, Hans Adler, Humanengr, Neuralwarp, Baudway,FactChecker1199, Kal-El-Bot, Subversive.sound, Tanhabot, Glane23, PV=nRT, Meisam, Legobot, Luckas-bot, Yobot, Ash4Math, Shva-habi, Omnipaedista, RibotBOT, Thehelpfulbot, FrescoBot, MarcelB612, CodeBlock, MastiBot, FoxBot, Duoduoduo, Xnn, EmausBot,Hikaslap, TuHan-Bot, Cobaltcigs, Wikfr, Karthikndr, Anita5192, Wcherowi, Widr, Strike Eagle, PhnomPencil, Knwlgc, Dhoke san-ket, Victor Yus, Cerabot~enwiki, JPaestpreornJeolhlna, Yardimsever, CasaNostra, KoriganStone, Whamsicore, JMP EAX, Sweepy andAnonymous: 89

• Bijection, injection and surjection Source: https://en.wikipedia.org/wiki/Bijection%2C_injection_and_surjection?oldid=670091181Contributors: The Anome, TakuyaMurata, PingPongBoy, Revolver, Charles Matthews, Altenmann, Mattflaschen, Tobias Bergemann,Rock69~enwiki, Antonis Christofides, Giftlite, MathKnight, Jcw69, Rich Farmbrough, Luqui, Bender235, Blotwell, Oleg Alexandrov,Ryan Reich, Jacj, Dpr, MarSch, Salix alba, FlaBot, Mathbot, YurikBot, Hairy Dude, Conscious, KSmrq, Grubber, Zwobot, Cconnett,Mastercampbell, InverseHypercube, XudongGuan~enwiki, Da nuke, Lambiam, Mike Fikes, CBM, Gregbard, Jac16888, Barticus88, Kon-radek, Headbomb, NLuchs, Magioladitis, Quantling, Perel, Versus22, Eraheem, Addbot, MrOllie, Jarble, AnomieBOT, Materialscien-tist, DannyAsher, Omnipaedista, Sławomir Biały, Oxonienses, Dmwpowers, Wikfr, Anita5192, ClueBot NG, Wcherowi, TricksterWolf,Jochen Burghardt, CSB radio, AwesomeEvilGenius, Sphinx-muse, Jascharnhorst, Nbro and Anonymous: 54

• Binary relation Source: https://en.wikipedia.org/wiki/Binary_relation?oldid=677636914 Contributors: AxelBoldt, Bryan Derksen, Zun-dark, Tarquin, Jan Hidders, Roadrunner, Mjb, Tomo, Patrick, Xavic69, Michael Hardy, Wshun, Isomorphic, Dominus, Ixfd64, Takuya-Murata, Charles Matthews, Timwi, Dcoetzee, Jitse Niesen, Robbot, Chocolateboy, MathMartin, Tobias Bergemann, Giftlite, Fropuff,Dratman, Jorge Stolfi, Jlr~enwiki, Andycjp, Quarl, Guanabot, Yuval madar, Slipstream, Paul August, Elwikipedista~enwiki, Shanes,EmilJ, Randall Holmes, Ardric47, Obradovic Goran, Eje211, Alansohn, Dallashan~enwiki, Keenan Pepper, PAR, Adrian.benko, OlegAlexandrov, Joriki, Linas, MFH, Dpv, Pigcatian, Penumbra2000, Fresheneesz, Chobot, YurikBot, Hairy Dude, Koffieyahoo, Trovatore,Bota47, Arthur Rubin, Netrapt, SmackBot, Royalguard11, SEIBasaurus, Cybercobra, Jon Awbrey, Turms, Lambiam, Dbtfz, Mr Stephen,Mets501, Dreftymac, Happy-melon, Petr Matas, CRGreathouse, CBM, Yrodro, WillowW, Xantharius, Thijs!bot, Egriffin, Rlupsa, JAnD-bot, MER-C, Magioladitis, Vanish2, David Eppstein, Robin S, Akurn, Adavidb, LajujKej, Owlgorithm, Djjrjr, Policron, DavidCBryant,Quux0r, VolkovBot, Boute, Vipinhari, Anonymous Dissident, PaulTanenbaum, Jackfork, Wykypydya, Dmcq, AlleborgoBot, AHMartin,Ocsenave, Sftd, Paradoctor, Henry Delforn (old), MiNombreDeGuerra, DuaneLAnderson, Anchor Link Bot, CBM2, Classicalecon,ClueBot, Snigbrook, Rhubbarb, Hans Adler, SilvonenBot, BYS2, Plmday, Addbot, LinkFA-Bot, Tide rolls, Jarble, Legobot, Luckas-bot,Yobot, Ht686rg90, Pcap, Labus, Nallimbot, Reindra, FredrikMeyer, AnomieBOT, Floquenbeam, Royote, Hahahaha4, Materialscientist,

https://en.wikipedia.org/wiki/Accessibility_relation?oldid=664002580

https://en.wikipedia.org/wiki/Allegory_(category_theory)?oldid=636236531

https://en.wikipedia.org/wiki/Alternatization?oldid=674572806

https://en.wikipedia.org/wiki/Ancestral_relation?oldid=668533172

https://en.wikipedia.org/wiki/Antisymmetric_relation?oldid=653233446

https://en.wikipedia.org/wiki/Asymmetric_relation?oldid=653233105

https://en.wikipedia.org/wiki/Better-quasi-ordering?oldid=640379072

https://en.wikipedia.org/wiki/Bidirectional_transformation?oldid=646599869

https://en.wikipedia.org/wiki/Bijection?oldid=670090850

https://en.wikipedia.org/wiki/Bijection%252C_injection_and_surjection?oldid=670091181

https://en.wikipedia.org/wiki/Binary_relation?oldid=677636914

78.10. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 287

Belkovich, Citation bot, Racconish, Jellystones, Xqbot, Isheden, Geero, GhalyBot, Ernsts, Howard McCay, Constructive editor, Mark Re-nier, Mfwitten, RandomDSdevel, NearSetAccount, SpaceFlight89, Yunshui, Miracle Pen, Brambleclawx, RjwilmsiBot, Nomen4Omen,Chharvey, SporkBot, OnePt618, Sameer143, Socialservice, ResearchRave, ClueBot NG, Wcherowi, Frietjes, Helpful Pixie Bot, Ko-ertefa, ChrisGualtieri, YFdyh-bot, Dexbot, Makecat-bot, Lerutit, Jochen Burghardt, Jodosma, Karim132, Monkbot, Pratincola, ,Some1Redirects4You and Anonymous: 102

• Cointerpretability Source: https://en.wikipedia.org/wiki/Cointerpretability?oldid=612104861 Contributors: Charles Matthews, Dys-prosia, Kntg, PWilkinson, Aleph0~enwiki, Oleg Alexandrov, Mathbot, Gregbard, David Eppstein, Hans Adler and Gamewizard71

• Commutative property Source: https://en.wikipedia.org/wiki/Commutative_property?oldid=672555787 Contributors: AxelBoldt, Zun-dark, Tarquin, Andre Engels, Christian List, Toby~enwiki, Toby Bartels, Patrick, Michael Hardy, Wshun, Ixfd64, GTBacchus, Ahoerste-meier, Snoyes, Jll, Pizza Puzzle, Ideyal, Charles Matthews, Wikiborg, Dysprosia, Jeffq, Robbot, RedWolf, Romanm, Robinh, Isopropyl,Tobias Bergemann, Enochlau, Giftlite, BenFrantzDale, Lupin, Herbee, Peruvianllama, Waltpohl, Frencheigh, Gdr, Knutux, OverlordQ,B.d.mills, Chris Howard, Mormegil, Rich Farmbrough, Ebelular, Mikael Brockman, Dbachmann, Paul August, MisterSheik, El C, Szquir-rel, Touriste, Samadam, Malcolm rowe, Jumbuck, Arthena, Mattpickman, Mlessard, Burn, Mlm42, Stillnotelf, Tony Sidaway, OlegAlexandrov, The JPS, Linas, Justinlebar, Jeff3000, Palica, Ashmoo, Graham87, Josh Parris, Rjwilmsi, Salix alba, Vegaswikian, FlaBot,VKokielov, Ground Zero, Srleffler, Masnevets, Reetep, YurikBot, Hairy Dude, Wolfmankurd, Michael Slone, Rick Norwood, SamuelHuang, Derek.cashman, FF2010, Petri Krohn, Vicarious, SmackBot, YellowMonkey, Slashme, Melchoir, KocjoBot~enwiki, Jab843,PJTraill, Chris the speller, Bluebot, Master of Puppets, Thumperward, SchfiftyThree, Complexica, Octahedron80, DHN-bot~enwiki, Jus-tUser, Cybercobra, Wybot, Thehakimboy, Acdx, Bando26, 16@r, Childzy, Dan Gluck, Iridescent, Dreftymac, DBooth, Robert.McGibbon,Floridi~enwiki, Unmitigated Success, Gregbard, MichaelRWolf, Cydebot, Larsnostdal, Kozuch, JamesAM, Headbomb, Second Quanti-zation, Wmasterj, Thomprod, Dzer0, Grayshi, Escarbot, Fr33ke, AntiVandalBot, Nacho Librarian, Gcm, 100110100, Mikemill, Wikidude-man, DAGwyn, Dirac66, JoergenB, MartinBot, Nev1, Daniele.tampieri, Haseldon, Policron, KylieTastic, Sarregouset, Useight, VolkovBot,TreasuryTag, Am Fiosaigear~enwiki, Philip Trueman, TXiKiBoT, Anonymous Dissident, Aaron Rotenberg, Geometry guy, Spinningspark,Life, Liberty, Property, SieBot, Ivan Štambuk, Legion fi, Toddst1, Flyer22, Xvani, Weston.pace, OKBot, Mike2vil, Francvs, Classi-calecon, ClueBot, Cliff, Bloodholds, R000t, CounterVandalismBot, Deathnomad, Excirial, Ftbhrygvn, Joe8824, Nafis ru, Stephen Pop-pitt, Addbot, LaaknorBot, SpBot, Gail, Luckas-bot, Yobot, Weisicong, DemocraticLuntz, Citation bot, MauritsBot, Xqbot, Dithridge,12cookk, Ubcule, GrouchoBot, Omnipaedista, Dger, HamburgerRadio, MacMed, Pinethicket, Adlerbot, Psimmler, ThinkEnemies, JVSmithy, Onel5969, Ujoimro, Aceshooter, Slightsmile, Quondum, Joshlepaknpsa, Wayne Slam, Arnaugir, Scientific29, ClueBot NG,Wcherowi, Gilderien, Sayginer, Marechal Ney, Widr, AvocatoBot, Mark Arsten, Ameulen11, CeraBot, ChrisGualtieri, None but shininghours, Khazar2, Dexbot, Stephan Kulla, Fox2k11, Dskjhgds, DavidLeighEllis, Davidliu0421, Wikibritannica, Niallhoranluv123, JMPEAX, Troolium, Hinmatóowyalahtqit, Holt Mcdougal, ABCDEFAD, Fazbear7891 and Anonymous: 180

• Comparability Source: https://en.wikipedia.org/wiki/Comparability?oldid=660335216 Contributors: Patrick, Charles Matthews, To-bias Bergemann, Tsirel, Oleg Alexandrov, Rjwilmsi, Archelon, 16@r, Mets501, CharacterZero, ThreeBlindMice, THF, David Eppstein,PaulTanenbaum, Burdel británico, Justin W Smith, SchreiberBike, Yobot, Gamewizard71, ChuispastonBot, Snotbot and Anonymous: 3

• Composition of relations Source: https://en.wikipedia.org/wiki/Composition_of_relations?oldid=644804909 Contributors: Rp, AugPi,Charles Matthews, Giftlite, EmilJ, Oliphaunt, MFH, SixWingedSeraph, MarSch, Nbarth, Lambiam, Happy-melon, CBM, Sam Staton,David Eppstein, Synthebot, Classicalecon, Hans Adler, Addbot, Luckas-bot, Yobot, Pcap, FrescoBot, Gamewizard71, Quondum, AgileAntechinus, JMP EAX and Anonymous: 6

• Congruence relation Source: https://en.wikipedia.org/wiki/Congruence_relation?oldid=655805590Contributors: AxelBoldt, Toby~enwiki,Toby Bartels, Michael Hardy, Charles Matthews, Jitse Niesen, Greenrd, Aleph4, Romanm, MathMartin, Henrygb, Tosha, Giftlite, Arved,Waltpohl, Mani1, ZeroOne, Boger1, PWilkinson, WojciechSwiderski~enwiki, Bookandcoffee, Oleg Alexandrov, Bluemoose, Marudub-shinki, Pako, Pasky, Kevmitch, DavidHouse~enwiki, Reyk, SmackBot, Imz, Mgreenbe, BiT, Mhss, Bluebot, Mohamed Al-Dabbagh,Jim.belk, Mets501, Vaughan Pratt, CRGreathouse, CBM, Sam Staton, Goldencako, Thijs!bot, Gdickeson, JAnDbot, Magioladitis, Jo-ergenB, VolkovBot, EuTuga, Anchor Link Bot, Amahoney, Sandeepjshenoy, Hans Adler, Palnot, Addbot, Mancini0, Legobot, Yobot,Calle, DannyAsher, Obersachsebot, GrouchoBot, FrescoBot, Stpasta, TobeBot, Dinamik-bot, Ayamewolfe, EmausBot, ZéroBot, ToshioYamaguchi, Elaz85, MerlIwBot, SteenthIWbot, Pietro13 and Anonymous: 30

• Contour set Source: https://en.wikipedia.org/wiki/Contour_set?oldid=576581920 Contributors: Giftlite, John Quiggin, Ms2ger, CBM,WillowW, Arch dude, David Eppstein, SlamDiego, RockMFR, Addbot, Xp54321, Econotechie, AndersBot, Flewis, Kornsystem69,WaysToEscape, Neil P. Quinn, MahdiBot, Yamaha5 and Anonymous: 1

• Coreflexive relation Source: https://en.wikipedia.org/wiki/Coreflexive_relation?oldid=316718782 Contributors: Michael Hardy, Xez-beth, H8erade, Ctylikkehl, Kms15 and Anonymous: 1

• Demonic composition Source: https://en.wikipedia.org/wiki/Demonic_composition?oldid=633771432 Contributors: Michael Hardy,David Eppstein, LokiClock, Classicalecon, AnomieBOT and Anonymous: 2

• Dense order Source: https://en.wikipedia.org/wiki/Dense_order?oldid=635100176 Contributors: EmilJ, Physicistjedi, MarSch, MichaelSlone, SmackBot, Imz, Melchoir, Turms, JAnDbot, VolkovBot, TXiKiBoT, Palnot, Addbot, ב ,.דניאל Pcap, Erik9bot, ZéroBot, HelpfulPixie Bot, Qetuth, Brirush and Anonymous: 6

• Dependence relation Source: https://en.wikipedia.org/wiki/Dependence_relation?oldid=668236798Contributors: Michael Hardy, CharlesMatthews, Jitse Niesen, Josh Parris, Wavelength, Robbjedi, Keegan, Geometry guy and 777sms

• Dependency relation Source: https://en.wikipedia.org/wiki/Dependency_relation?oldid=668236930Contributors: Michael Hardy, WilliamM. Connolley, GPHemsley, Robbot, Wizzy, Goochelaar, Linas, Jsnx, SmackBot, Chris the speller, NickPenguin, Mukake, David Eppstein,Homei, Classicalecon, Addbot, Yobot, WikitanvirBot, Mark viking, W. P. Uzer, Christian Nassif-Haynes, JMP EAX and Anonymous: 4

• Difunctional Source: https://en.wikipedia.org/wiki/Binary_relation?oldid=677636914 Contributors: AxelBoldt, Bryan Derksen, Zun-dark, Tarquin, Jan Hidders, Roadrunner, Mjb, Tomo, Patrick, Xavic69, Michael Hardy, Wshun, Isomorphic, Dominus, Ixfd64, Takuya-Murata, Charles Matthews, Timwi, Dcoetzee, Jitse Niesen, Robbot, Chocolateboy, MathMartin, Tobias Bergemann, Giftlite, Fropuff,Dratman, Jorge Stolfi, Jlr~enwiki, Andycjp, Quarl, Guanabot, Yuval madar, Slipstream, Paul August, Elwikipedista~enwiki, Shanes,EmilJ, Randall Holmes, Ardric47, Obradovic Goran, Eje211, Alansohn, Dallashan~enwiki, Keenan Pepper, PAR, Adrian.benko, OlegAlexandrov, Joriki, Linas, MFH, Dpv, Pigcatian, Penumbra2000, Fresheneesz, Chobot, YurikBot, Hairy Dude, Koffieyahoo, Trovatore,Bota47, Arthur Rubin, Netrapt, SmackBot, Royalguard11, SEIBasaurus, Cybercobra, Jon Awbrey, Turms, Lambiam, Dbtfz, Mr Stephen,Mets501, Dreftymac, Happy-melon, Petr Matas, CRGreathouse, CBM, Yrodro, WillowW, Xantharius, Thijs!bot, Egriffin, Rlupsa, JAnD-bot, MER-C, Magioladitis, Vanish2, David Eppstein, Robin S, Akurn, Adavidb, LajujKej, Owlgorithm, Djjrjr, Policron, DavidCBryant,

https://en.wikipedia.org/wiki/Cointerpretability?oldid=612104861

https://en.wikipedia.org/wiki/Commutative_property?oldid=672555787

https://en.wikipedia.org/wiki/Comparability?oldid=660335216

https://en.wikipedia.org/wiki/Composition_of_relations?oldid=644804909

https://en.wikipedia.org/wiki/Congruence_relation?oldid=655805590

https://en.wikipedia.org/wiki/Contour_set?oldid=576581920

https://en.wikipedia.org/wiki/Coreflexive_relation?oldid=316718782

https://en.wikipedia.org/wiki/Demonic_composition?oldid=633771432

https://en.wikipedia.org/wiki/Dense_order?oldid=635100176

https://en.wikipedia.org/wiki/Dependence_relation?oldid=668236798

https://en.wikipedia.org/wiki/Dependency_relation?oldid=668236930



Quux0r, VolkovBot, Boute, Vipinhari, Anonymous Dissident, PaulTanenbaum, Jackfork, Wykypydya, Dmcq, AlleborgoBot, AHMartin,Ocsenave, Sftd, Paradoctor, Henry Delforn (old), MiNombreDeGuerra, DuaneLAnderson, Anchor Link Bot, CBM2, Classicalecon,ClueBot, Snigbrook, Rhubbarb, Hans Adler, SilvonenBot, BYS2, Plmday, Addbot, LinkFA-Bot, Tide rolls, Jarble, Legobot, Luckas-bot,Yobot, Ht686rg90, Pcap, Labus, Nallimbot, Reindra, FredrikMeyer, AnomieBOT, Floquenbeam, Royote, Hahahaha4, Materialscientist,Belkovich, Citation bot, Racconish, Jellystones, Xqbot, Isheden, Geero, GhalyBot, Ernsts, Howard McCay, Constructive editor, Mark Re-nier, Mfwitten, RandomDSdevel, NearSetAccount, SpaceFlight89, Yunshui, Miracle Pen, Brambleclawx, RjwilmsiBot, Nomen4Omen,Chharvey, SporkBot, OnePt618, Sameer143, Socialservice, ResearchRave, ClueBot NG, Wcherowi, Frietjes, Helpful Pixie Bot, Ko-ertefa, ChrisGualtieri, YFdyh-bot, Dexbot, Makecat-bot, Lerutit, Jochen Burghardt, Jodosma, Karim132, Monkbot, Pratincola, ,Some1Redirects4You and Anonymous: 102

• Directed set Source: https://en.wikipedia.org/wiki/Directed_set?oldid=663876888Contributors: AxelBoldt, The Anome, SimonP, Patrick,Michael Hardy, AugPi, Nikai, Revolver, Dfeuer, Dysprosia, Tobias Bergemann, Giftlite, Markus Krötzsch, Smimram, Paul August,Varuna, Msh210, Eric Kvaalen, SteinbDJ, Linas, Dionyziz, Salix alba, Margosbot~enwiki, Hairy Dude, Zwobot, Futanari, Reedy, Fitch,Mhss, Vaughan Pratt, CBM, Blaisorblade, Larvy, JoergenB, CommonsDelinker, LokiClock, Don4of4, AlleborgoBot, SieBot, Tom Le-inster, He7d3r, Beroal, Palnot, Plmday, Legobot, Luckas-bot, Obersachsebot, Yaddie, ZéroBot, Haraldbre, Helpful Pixie Bot, Freeze S,Jochen Burghardt, Zoydb, Austrartsua, Some1Redirects4You and Anonymous: 25

• Equality (mathematics) Source: https://en.wikipedia.org/wiki/Equality_(mathematics)?oldid=672048328 Contributors: Toby Bartels,Patrick, Michael Hardy, TakuyaMurata, Looxix~enwiki, Pizza Puzzle, Charles Matthews, Dysprosia, WhisperToMe, Banno, Robbot,RedWolf, Lowellian, Tobias Bergemann, Alan Liefting, Giftlite, Christopher Parham, Recentchanges, Michael Devore, DefLog~enwiki,Chowbok, Smiller933, Shahab, AlexG, Wrp103, Plugwash, Rgdboer, Spoon!, Iltseng, PWilkinson, MPerel, Jumbuck, Msh210, Hu,Japanese Searobin, Simetrical, Linas, MattGiuca, Isnow, Wbeek, Qwertyus, Island, Scottkeir, Jshadias, Pasky, FlaBot, VKokielov, Mar-gosbot~enwiki, Fresheneesz, Chobot, DVdm, Gwernol, Laurentius, Piet Delport, Pnrj, TransUtopian, Reyk, Tinlv7, SmackBot, RDBury,Incnis Mrsi, Melchoir, Blue520, Josephprymak, BiT, Bluebot, Nbarth, Jdthood, Jon Awbrey, Lambiam, Attys, Loadmaster, Mets501,Tauʻolunga, CBM, Sdorrance, Simeon, Gregbard, Cydebot, Benzi455, Blaisorblade, Xantharius, Uv~enwiki, Cj67, Dugwiki, Anti-VandalBot, Malcolm, JAnDbot, Thenub314, Edward321, R'n'B, Anonymous Dissident, PaulTanenbaum, UnitedStatesian, Enigmaman,Vikrant42, Tachikoma’s All Memory, Flyer22, Ctxppc, ClueBot, BodhisattvaBot, SilvonenBot, Addbot, Debresser, Numbo3-bot, Apteva,Legobot, Luckas-bot, Yobot, TaBOT-zerem, Amirobot, Pcap, KamikazeBot, Ningauble, Bryan.burgers, MassimoAr, AnomieBOT, King-pin13, Citation bot, Capricorn42, Kevfest08, NOrbeck, VladimirReshetnikov, Der Falke, FrescoBot, Tkuvho, AmphBot, RedBot, Jauhienij,TobeBot, Belovedeagle, Vrenator, CobraBot, Duoduoduo, Ebe123, ZéroBot, Sungzungkim, D.Lazard, ClueBot NG, Iiii I I I, Wcherowi,Faus, ChrisGualtieri, Brirush, DialaceStarvy, Monkbot, Lizard Pancakes123456789012345678901234567890, Gmalaven, This is a mo-bile phone and Anonymous: 75

• Equipollence (geometry) Source: https://en.wikipedia.org/wiki/Equipollence_(geometry)?oldid=674436874Contributors: Michael Hardy,Mdob, Rgdboer, Siddhant, Sadads, Addbot, Omnipaedista, Erik9bot, J.Victor, Specs112, Makhokh and Brad7777

• Equivalence class Source: https://en.wikipedia.org/wiki/Equivalence_class?oldid=667055947Contributors: AxelBoldt, Zundark, Patrick,Michael Hardy, Wshun, Salsa Shark, Revolver, Charles Matthews, Dysprosia, Wolfgang Kufner, Greenrd, Hyacinth, Psychonaut, Naddy,GreatWhiteNortherner, Tobias Bergemann, Giftlite, WiseWoman, Lethe, Fropuff, Fuzzy Logic, Noisy, Tibbetts, Liuyao, Rgdboer,Msh210, MattGiuca, Graham87, Salix alba, Mike Segal, Jameshfisher, Laurentius, Hede2000, Arthur Rubin, Lunch, SmackBot, Mhss,Nbarth, Javalenok, Lhf, Mets501, Andrew Delong, Egriffin, Magioladitis, David Eppstein, VolkovBot, LokiClock, Rjgodoy, Quietbri-tishjim, Dogah, Henry Delforn (old), Sjn28, Classicalecon, Watchduck, Kausikghatak, Addbot, WikiDreamer Bot, Calle, Rinke 80,Erik9bot, HJ Mitchell, WillNess, Igor Yalovecky, Quondum, D.Lazard, Herebo, Wcherowi, Rpglover64, ChrisGualtieri, Brirush, Markviking, A4b3c2d1e0f, Riddleh, Verdana Bold, Addoergosum and Anonymous: 45

• Equivalence relation Source: https://en.wikipedia.org/wiki/Equivalence_relation?oldid=672050542 Contributors: AxelBoldt, Zundark,Toby Bartels, PierreAbbat, Ryguasu, Stevertigo, Patrick, Michael Hardy, Wshun, Dominus, TakuyaMurata, William M. Connolley, AugPi,Silverfish, Ideyal, Revolver, Charles Matthews, Dysprosia, Hyacinth, Fibonacci, Phys, McKay, GPHemsley, Robbot, Fredrik, Romanm,COGDEN, Ashley Y, Bkell, Tobias Bergemann, Tosha, Giftlite, Arved, ShaunMacPherson, Lethe, Herbee, Fropuff, LiDaobing, AlexG,Paul August, Elwikipedista~enwiki, FirstPrinciples, Rgdboer, Spearhead, Smalljim, SpeedyGonsales, Obradovic Goran, Haham hanuka,Kierano, Msh210, Keenan Pepper, PAR, Jopxton, Oleg Alexandrov, Linas, MFH, BD2412, Salix alba, [email protected], MarkJ, Epitome83, Chobot, Algebraist, Roboto de Ajvol, YurikBot, Wavelength, RussBot, Nils Grimsmo, BOT-Superzerocool, Googl, Larry-LACa, Arthur Rubin, Pred, Cjfsyntropy, Draicone, RonnieBrown, SmackBot, Adam majewski, Melchoir, Stifle, Srnec, Gilliam, Kurykh,Concerned cynic, Foxjwill, Vanished User 0001, Michael Ross, Jon Awbrey, Jim.belk, Feraudyh, CredoFromStart, Michael Kinyon,JHunterJ, Vanished user 8ij3r8jwefi, Mets501, Rschwieb, Captain Wacky, JForget, CRGreathouse, CBM, 345Kai, Gregbard, Doctor-matt, PepijnvdG, Tawkerbot4, Xantharius, Hanche, BetacommandBot, Thijs!bot, Egriffin, Rlupsa, WilliamH, Rnealh, Salgueiro~enwiki,JAnDbot, Thenub314, Magioladitis, VoABot II, JamesBWatson, MetsBot, Robin S, Philippe.beaudoin, Pekaje, Pomte, Interwal, Cpiral,GaborLajos, Policron, Taifunbrowser, Idioma-bot, Station1, Davehi1, Billinghurst, Geanixx, AlleborgoBot, SieBot, BotMultichill, This,that and the other, Henry Delforn (old), Aspects, OKBot, Bulkroosh, C1wang, Classicalecon, Wmli, Kclchan, Watchduck, Hans Adler,Qwfp, Cdegremo, Palnot, XLinkBot, Gerhardvalentin, Libcub, LaaknorBot, CarsracBot, Dyaa, Legobot, Luckas-bot, Yobot, Ht686rg90,Gyro Copter, Andy.melnikov, ArthurBot, Xqbot, GrouchoBot, Lenore, RibotBOT, Antares5245, Sokbot3000, Anthonystevens2, ARan-domNicole, Tkuvho, SpaceFlight89, TobeBot, Miracle Pen, EmausBot, ReneGMata, AvicBot, Vanished user fois8fhow3iqf9hsrlgkjw4tus,TyA, Donner60, Gottlob Gödel, ClueBot NG, Bethre, Helpful Pixie Bot, Mark Arsten, ChrisGualtieri, Rectipaedia, YFdyh-bot, Noix07,Adammwagner, Damonamc and Anonymous: 108

• Euclidean relation Source: https://en.wikipedia.org/wiki/Euclidean_relation?oldid=557928957 Contributors: Toby Bartels, Giftlite,EmilJ, PAR, Salix alba, Lhf, Turms, Gregbard, Egriffin, David Eppstein, Robertgreer, Cdegremo, Yangtseyangtse, Helpful Pixie Botand Anonymous: 4

• Exceptional isomorphism Source: https://en.wikipedia.org/wiki/Exceptional_isomorphism?oldid=629067192 Contributors: MichaelHardy, Charles Matthews, Tobias Bergemann, Rjwilmsi, Koavf, Wavelength, SmackBot, Nbarth, Tamfang, David Eppstein, Citation bot,Twri, Jamontaldi, Teddyktchan and Anonymous: 5

• Fiber (mathematics) Source: https://en.wikipedia.org/wiki/Fiber_(mathematics)?oldid=638573433Contributors: Chinju, Charles Matthews,Oleg Alexandrov, Christopher Thomas, MarSch, LkNsngth, Jon Awbrey, Krasnoludek, JRSpriggs, CBM, Kilva, OrenBochman, Camrn86,LokiClock, Dmcq, JP.Martin-Flatin, Addbot, Ptbotgourou, Ciphers, Erik9bot, Artem M. Pelenitsyn, ZéroBot, Beaumont877, Qetuth,SillyBunnies and Anonymous: 8

https://en.wikipedia.org/wiki/Directed_set?oldid=663876888

https://en.wikipedia.org/wiki/Equality_(mathematics)?oldid=672048328

https://en.wikipedia.org/wiki/Equipollence_(geometry)?oldid=674436874

https://en.wikipedia.org/wiki/Equivalence_class?oldid=667055947

https://en.wikipedia.org/wiki/Equivalence_relation?oldid=672050542

https://en.wikipedia.org/wiki/Euclidean_relation?oldid=557928957

https://en.wikipedia.org/wiki/Exceptional_isomorphism?oldid=629067192

https://en.wikipedia.org/wiki/Fiber_(mathematics)?oldid=638573433


• Finitary relation Source: https://en.wikipedia.org/wiki/Finitary_relation?oldid=674490128 Contributors: Damian Yerrick, AxelBoldt,The Anome, Tarquin, Jan Hidders, Patrick, Michael Hardy, Wshun, Kku, Ellywa, Andres, Charles Matthews, Dcoetzee, Hyacinth, Rob-bot, Romanm, MathMartin, Tobias Bergemann, Alan Liefting, Marc Venot, Giftlite, Almit39, Zfr, Starx, PhotoBox, Erc, ArnoldRein-hold, Paul August, Elwikipedista~enwiki, Randall Holmes, Obradovic Goran, Oleg Alexandrov, Woohookitty, Mangojuice, MichielHelvensteijn, Isnow, Qwertyus, Dpr, MarSch, Salix alba, Oblivious, Mathbot, Jrtayloriv, Chobot, YurikBot, Hairy Dude, Dmharvey,RussBot, Muu-karhu, Zwobot, Bota47, Arthur Rubin, Reyk, Netrapt, Claygate, JoanneB, Pred, GrinBot~enwiki, SmackBot, Mmernex,Unyoyega, Nbarth, DHN-bot~enwiki, Tinctorius, Jon Awbrey, Henning Makholm, Lambiam, Dfass, Newone, Aeons, CRGreathouse,Gregbard, King Bee, Kilva, Escarbot, Salgueiro~enwiki, Nosbig, JAnDbot, .anacondabot, Tarif Ezaz, VoABot II, Jonny Cache, Der-Hexer, Mike.lifeguard, And Dedicated To, Aervanath, VolkovBot, Rponamgi, The Tetrast, Mscman513, GirasoleDE, Newbyguesses,SieBot, Phe-bot, Paolo.dL, Siorale, Skeptical scientist, Sheez Louise, Mild Bill Hiccup, DragonBot, Cenarium, Palnot, Cat Dancer WS,Kal-El-Bot, Addbot, MrOllie, ChenzwBot, Ariel Black, SpBot, Yobot, Ptbotgourou, Bgttgb, QueenCake, Dinnertimeok, AnomieBOT,JRB-Europe, Xqbot, Nishantjr, Howard McCay, Paine Ellsworth, Throw it in the Fire, RandomDSdevel, Miracle Pen, Straightontillmorn-ing, ZéroBot, Cackleberry Airman, Paulmiko, Tijfo098, Mister Stan, Deer*lake, Frietjes, ChrisGualtieri, Fuebar, Brirush, Mark viking,Andrei Petre, KasparBot, Some1Redirects4You and Anonymous: 52

• Foundational relation Source: https://en.wikipedia.org/wiki/Foundational_relation?oldid=659203908 Contributors: Michael Hardy,BiH and IkamusumeFan

• Homogeneous relation Source: https://en.wikipedia.org/wiki/Binary_relation?oldid=677636914 Contributors: AxelBoldt, Bryan Derk-sen, Zundark, Tarquin, Jan Hidders, Roadrunner, Mjb, Tomo, Patrick, Xavic69, Michael Hardy, Wshun, Isomorphic, Dominus, Ixfd64,TakuyaMurata, Charles Matthews, Timwi, Dcoetzee, Jitse Niesen, Robbot, Chocolateboy, MathMartin, Tobias Bergemann, Giftlite, Frop-uff, Dratman, Jorge Stolfi, Jlr~enwiki, Andycjp, Quarl, Guanabot, Yuval madar, Slipstream, Paul August, Elwikipedista~enwiki, Shanes,EmilJ, Randall Holmes, Ardric47, Obradovic Goran, Eje211, Alansohn, Dallashan~enwiki, Keenan Pepper, PAR, Adrian.benko, OlegAlexandrov, Joriki, Linas, MFH, Dpv, Pigcatian, Penumbra2000, Fresheneesz, Chobot, YurikBot, Hairy Dude, Koffieyahoo, Trovatore,Bota47, Arthur Rubin, Netrapt, SmackBot, Royalguard11, SEIBasaurus, Cybercobra, Jon Awbrey, Turms, Lambiam, Dbtfz, Mr Stephen,Mets501, Dreftymac, Happy-melon, Petr Matas, CRGreathouse, CBM, Yrodro, WillowW, Xantharius, Thijs!bot, Egriffin, Rlupsa, JAnD-bot, MER-C, Magioladitis, Vanish2, David Eppstein, Robin S, Akurn, Adavidb, LajujKej, Owlgorithm, Djjrjr, Policron, DavidCBryant,Quux0r, VolkovBot, Boute, Vipinhari, Anonymous Dissident, PaulTanenbaum, Jackfork, Wykypydya, Dmcq, AlleborgoBot, AHMartin,Ocsenave, Sftd, Paradoctor, Henry Delforn (old), MiNombreDeGuerra, DuaneLAnderson, Anchor Link Bot, CBM2, Classicalecon,ClueBot, Snigbrook, Rhubbarb, Hans Adler, SilvonenBot, BYS2, Plmday, Addbot, LinkFA-Bot, Tide rolls, Jarble, Legobot, Luckas-bot,Yobot, Ht686rg90, Pcap, Labus, Nallimbot, Reindra, FredrikMeyer, AnomieBOT, Floquenbeam, Royote, Hahahaha4, Materialscientist,Belkovich, Citation bot, Racconish, Jellystones, Xqbot, Isheden, Geero, GhalyBot, Ernsts, Howard McCay, Constructive editor, Mark Re-nier, Mfwitten, RandomDSdevel, NearSetAccount, SpaceFlight89, Yunshui, Miracle Pen, Brambleclawx, RjwilmsiBot, Nomen4Omen,Chharvey, SporkBot, OnePt618, Sameer143, Socialservice, ResearchRave, ClueBot NG, Wcherowi, Frietjes, Helpful Pixie Bot, Ko-ertefa, ChrisGualtieri, YFdyh-bot, Dexbot, Makecat-bot, Lerutit, Jochen Burghardt, Jodosma, Karim132, Monkbot, Pratincola, ,Some1Redirects4You and Anonymous: 102

• Hypostatic abstraction Source: https://en.wikipedia.org/wiki/Hypostatic_abstraction?oldid=582042811Contributors: Bevo, MisfitToys,El C, Diego Moya, Versageek, Jeffrey O. Gustafson, Magister Mathematicae, DoubleBlue, TeaDrinker, Brandmeister (old), Closedmouth,C.Fred, Rajah9, JasonMR, Jon Awbrey, Inhahe, JzG, Slakr, CBM, Gogo Dodo, Hut 8.5, Brigit Zilwaukee, Yolanda Zilwaukee, Karrade,Mike V, The Tetrast, Rjd0060, Wolf of the Steppes, Doubtentry, Icharus Ixion, Hans Adler, Buchanan’s Navy Sec, Mr. Peabody’s Boy,Overstay, Marsboat, Unco Guid, Viva La Information Revolution!, Autocratic Uzbek, Poke Salat Annie, Flower Mound Belle, NavyPierre, Mrs. Lovett’s Meat Puppets, Chester County Dude, Southeast Penna Poppa, Delaware Valley Girl, AnomieBOT, Paine Ellsworth,Gamewizard71, PhnomPencil and Anonymous: 3

• Idempotence Source: https://en.wikipedia.org/wiki/Idempotence?oldid=675763490 Contributors: Damian Yerrick, AxelBoldt, Zundark,Taw, Patrick, Chas zzz brown, Michael Hardy, Ahoerstemeier, Stevan White, Andres, Zarius, Revolver, Charles Matthews, Dysprosia, JitseNiesen, Glimz~enwiki, Hyacinth, Robbot, Mattblack82, Altenmann, Tobias Bergemann, Giftlite, Fropuff, Mboverload, DefLog~enwiki,Joseph Myers, Tothebarricades.tk, Andreas Kaufmann, Gcanyon, Mormegil, Rgdboer, Kwamikagami, Nigelj, Sleske, Obradovic Goran,Mdd, Abdulqabiz, Dirac1933, Bookandcoffee, Simetrical, Igny, MFH, Grammarbot, MarkHudson, Tokigun, FlaBot, Mathbot, Vonkje,DVdm, Roboto de Ajvol, Angus Lepper, RobotE, Hairy Dude, Michael Slone, Gaius Cornelius, FuzzyBSc, Aeusoes1, Deskana, Trova-tore, Sangwine, Sfnhltb, Crasshopper, Tomisti, Kompik, Evilbu, SmackBot, Elronxenu, Octahedron80, Nbarth, Syberghost, Cybercobra,Lambiam, Robofish, Loadmaster, JHunterJ, MedeaMelana, Rschwieb, Alex Selby, Dreftymac, Happy-melon, CRGreathouse, CmdrObot,Gregbard, Julian Mendez, Thijs!bot, Kilva, Konradek, DB Durham NC, RichardVeryard, RobHar, Gioto, Ouc, JAnDbot, Deflective, Ran-dal Oulton, Danculley, Vanish2, Albmont, [email protected], David Eppstein, Emw, Cander0000, Stolsvik, Catskineater, Robin S,Gwern, Bostonvaulter, TomyDuby, Krishnachandranvn, Policron, JohnBlackburne, TXiKiBoT, Una Smith, Broadbot, Jamelan, SieBot,LungZeno, Roujo, Svick, Stjarnblom, Niceguyedc, DragonBot, Alexbot, He7d3r, Herbert1000, Rswarbrick, Qwfp, Gniemeyer, Addbot,MrOllie, Download, Legobot, Luckas-bot, Yobot, AnomieBOT, Götz, JackieBot, NOrbeck, Kaoru Itou, RandomDSdevel, Pinethicket,Serols, Ms7821, Duoduoduo, Timtempleton, Peoplemerge, Wham Bam Rock II, Quondum, Dumbier, Deltahedron, Jochen Burghardt,00prometheus, JaconaFrere, Acominym, Loraof and Anonymous: 83

• Idempotent relation Source: https://en.wikipedia.org/wiki/Idempotent_relation?oldid=666347335Contributors: Michael Hardy, Bearcat,Deltahedron, Jochen Burghardt and Flokam

• Intransitivity Source: https://en.wikipedia.org/wiki/Intransitivity?oldid=676080474 Contributors: Michael Hardy, Booyabazooka, Rp,6birc, Radicalsubversiv, Andres, Charles Matthews, Ruakh, Giftlite, Andris, Quickwik, D6, Xezbeth, Paul August, Jnestorius, Spoon!,Polluks, SmackBot, Melchoir, Mauls, MisterHand, Lambiam, Iamagloworm, Dinkumator, CRGreathouse, CmdrObot, CBM, Thomas-meeks, Ael 2, Thijs!bot, VoABot II, Cnilep, Ddxc, Anchor Link Bot, PixelBot, DumZiBoT, Pa68, Addbot, Forich, WissensDürster,Undsoweiter, HRoestBot, ChronoKinetic, RjwilmsiBot, Chharvey, Diakov and Anonymous: 9

• Inverse relation Source: https://en.wikipedia.org/wiki/Inverse_relation?oldid=657149498 Contributors: Patrick, Charles Matthews, Al-tenmann, Tobias Bergemann, Giftlite, Kaldari, Spiffy sperry, El C, Versageek, Jeffrey O. Gustafson, MFH, YurikBot, Paul Erik, Knowled-geOfSelf, C.Fred, Nbarth, Faaaa, Nixeagle, Jon Awbrey, Mearnhardtfan, Coredesat, Slakr, DBooth, CBM, Thomasmeeks, Gogo Dodo,Mojo Hand, Hut 8.5, MetsBot, David Eppstein, Real World Apple, Ars Tottle, VolkovBot, Iamthedeus, Maelgwnbot, Classicalecon,Rjd0060, Overstay, Trainshift, Pluto Car, Unco Guid, Viva La Information Revolution!, Autocratic Uzbek, Poke Salat Annie, FlowerMound Belle, Navy Pierre, Mrs. Lovett’s Meat Puppets, Chester County Dude, Southeast Penna Poppa, Delaware Valley Girl, Myst-Bot, Omphaloskeptor, Addbot, Quercus solaris, Luckas-bot, Yobot, Pcap, Materialscientist, MathHisSci, ویکی ,علی John of Reading,SporkBot, Mark viking, JMP EAX and Anonymous: 12

https://en.wikipedia.org/wiki/Finitary_relation?oldid=674490128

https://en.wikipedia.org/wiki/Foundational_relation?oldid=659203908


https://en.wikipedia.org/wiki/Hypostatic_abstraction?oldid=582042811

https://en.wikipedia.org/wiki/Idempotence?oldid=675763490

https://en.wikipedia.org/wiki/Idempotent_relation?oldid=666347335

https://en.wikipedia.org/wiki/Intransitivity?oldid=676080474

https://en.wikipedia.org/wiki/Inverse_relation?oldid=657149498


• Inverse trigonometric functions Source: https://en.wikipedia.org/wiki/Inverse_trigonometric_functions?oldid=676298808 Contribu-tors: XJaM, Patrick, Michael Hardy, Stevenj, Dysprosia, Fibonacci, Robbot, Tobias Bergemann, Giftlite, Anville, SURIV, Daniel,levine,Pmanderson, Abdull, Discospinster, Osrevad, Zenohockey, Army1987, Alansohn, Anthony Appleyard, Wtmitchell, StradivariusTV, Ar-mando, Gerbrant, Emallove, R.e.b., Kri, Glenn L, Salvatore Ingala, Chobot, Visor, DVdm, Algebraist, YurikBot, Wavelength, Sceptre,Hede2000, KSmrq, Grafen, Int 80h, NorsemanII, Bamse, RDBury, Maksim-e~enwiki, Thelukeeffect, Eskimbot, Mhss, Mirokado, JC-Santos, PrimeHunter, Deathanatos, V1adis1av, Saippuakauppias, Lambiam, Eridani, Ian Vaughan, ChaoticLlama, CapitalR, JRSpriggs,Conrad.Irwin, HenningThielemann, Fommil, Rian.sanderson, Palmtree3000, Zalgo, Thijs!bot, Spikedmilk, Nonagonal Spider, EdJohn-ston, Luna Santin, Hannes Eder, JAnDbot, Ricardo sandoval, Jetstreamer, JNW, Albmont, Gamkiller, JoergenB, Ac44ck, Gwern,Isamil, Mythealias, Pomte, Knorlin, TungstenWolfram, Hennessey, Patrick, Bobianite, BentonMiller, Sigmundur, DavidCBryant, AlanU. Kennington, VolkovBot, Indubitably, LokiClock, Justtysen, VasilievVV, Riku92mr, Anonymous Dissident, Corvus coronoides, Dmcq,Vertciel, Logan, CagedKiller360, Aly89, AlanUS, ClueBot, JoeHillen, Bender2k14, Cenarium, Leandropls, Kiensvay, Nikhilkrgvr,Aaron north, Dthomsen8, DaL33T, Addbot, Fgnievinski, Iceblock, Zarcadia, Sleepaholic, Jasper Deng, Zorrobot, Luckas-bot, Yobot,Tohd8BohaithuGh1, Ptbotgourou, TaBOT-zerem, AnomieBOT, JackieBot, Nickweedon, Geek1337~enwiki, Diego Queiroz, Txebixev,St.nerol, Hdullin, GrouchoBot, Uniwersalista, SassoBot, Prari, Nixphoeni, D'ohBot, Kusluj, Emjayeff, Number Googol, Adammer-linsmith, TjBot, Jowa fan, EmausBot, ModWilson, Velowiki, X-4-V-I, Wham Bam Rock II, ZéroBot, Michael.YX.Wu, Isaac Euler,Tolly4bolly, Jay-Sebastos, Colin.campbell.27, Maschen, ChuispastonBot, Rmashhadi, ClueBot NG, Hdreuter, Helpful Pixie Bot, KLBot2,Vagobot, Crh23, YatharthROCK, StevinSimon, Tfr000, Modalanalytiker, Pratyya Ghosh, Ahmed Magdy Hosny, Brirush, Yardimsever,Wamiq, Jerming, Blackbombchu, Pqnlrn, DTL LAPOS, De Riban5, Monkbot, Arsenal CR7, Cdserio99 and Anonymous: 168

• Near sets Source: https://en.wikipedia.org/wiki/Near_sets?oldid=662493389 Contributors: Michael Hardy, Bearcat, Gandalf61, Giftlite,Rjwilmsi, Algebraist, SmackBot, MartinPoulter, NSH001, Sebras, Salih, Philip Trueman, JL-Bot, SchreiberBike, Tassedethe, Jarble,Alvin Seville, Erik9bot, FrescoBot, NSH002, NearSetAccount, AManWithNoPlan, Frietjes, Helpful Pixie Bot, BG19bot, Christopher-JamesHenry and Anonymous: 2

• Partial equivalence relation Source: https://en.wikipedia.org/wiki/Partial_equivalence_relation?oldid=672263741 Contributors: Zun-dark, Charles Matthews, Malcohol, Smimram, Rgdboer, Kierano, Cjoev, Alex Bakharev, AndrewWTaylor, SmackBot, Bluebot, Alansmithee, J. Finkelstein, Mets501, David Eppstein, R'n'B, Functor salad, Palnot, AnomieBOT, Erik9bot, Helpful Pixie Bot, Some1Redirects4Youand Anonymous: 3

• Partial function Source: https://en.wikipedia.org/wiki/Partial_function?oldid=670578722 Contributors: AxelBoldt, Jan Hidders, Ede-maine, Michael Hardy, Pizza Puzzle, Charles Matthews, Timwi, Zoicon5, Altenmann, MathMartin, Henrygb, Tobias Bergemann, Tosha,Connelly, Giftlite, Lethe, Waltpohl, DRE, Paul August, Rgdboer, Pearle, Sligocki, Schapel, Artur adib, LunaticFringe, Oleg Alexandrov,^demon, Mpatel, MFH, Salix alba, Dougluce, FlaBot, VKokielov, Jameshfisher, Vonkje, Hairy Dude, Grubber, Trovatore, Kooky, ArthurRubin, Crystallina, Jbalint, SmackBot, Incnis Mrsi, Reedy, Ppntori, Chris the speller, Bluebot, Tsca.bot, TheArchivist, Cícero, Wvbailey,Etatoby, CBM, Blaisorblade, Dvandersluis, Dricherby, Albmont, Timlevin, Ravikaushik, Joshua Issac, Dmcq, Paolo.dL, , Classicale-con, ClueBot, Gulmammad, Maheshexp, El bot de la dieta, Lightbot, Legobot, Luckas-bot, Yobot, Pcap, AnomieBOT, Isheden, Brook-swift, NoJr0xx, Nicolas Perrault III, SkpVwls, Unbitwise, AvicAWB, Quondum, This lousy T-shirt, YannickN, Jergas, Tomtom2357, SJDefender, JMP EAX, Shaelja and Anonymous: 41

• Partially ordered set Source: https://en.wikipedia.org/wiki/Partially_ordered_set?oldid=672760099 Contributors: Bryan Derksen, Zun-dark, Tomo, Patrick, Bcrowell, Chinju, TakuyaMurata, GTBacchus, AugPi, Charles Matthews, Timwi, Dcoetzee, Dysprosia, Doradus,Maximus Rex, Fibonacci, Tobias Bergemann, Giftlite, Markus Krötzsch, Fropuff, Peruvianllama, Jason Quinn, Neilc, Gubbubu, De-fLog~enwiki, MarkSweep, Urhixidur, TheJames, Paul August, Zaslav, Spoon!, Porton, Haham hanuka, DougOrleans, Msh210, OlegAlexandrov, Daira Hopwood, MFH, Salix alba, FlaBot, Vonkje, Chobot, Laurentius, Dmharvey, Vecter, JosephSilverman, Sanguinity,Modify, RDBury, Incnis Mrsi, Brick Thrower, Cesine, Zanetu, Jcarroll, Nbarth, Jdthood, Javalenok, Kjetil1001, Dreadstar, Esoth~enwiki,Mike Fikes, A. Pichler, Vaughan Pratt, CRGreathouse, L'œuf, CBM, Werratal, Rlupsa, CZeke, Ill logic, JAnDbot, MER-C, BrotherE,Tbleher, A3nm, David Eppstein, SlamDiego, Bissinger, Haseldon, Daniel5Ko, GaborLajos, NewEnglandYankee, Orphic, RobertDanielE-merson, TXiKiBoT, Digby Tantrum, PaulTanenbaum, Arcfrk, SieBot, Mochan Shrestha, TheGhostOfAdrianMineha, Thehotelambush,Megaloxantha, Peiresc~enwiki, Cheesefondue, Jludwig, ClueBot, Morinus, Justin W Smith, Methossant, Pi zero, Jonathanrcoxhead,Watchduck, ComputerGeezer, He7d3r, Hans Adler, Jtle515, Palnot, Marc van Leeuwen, Ankan babee, Addbot, Download, Luckyz,Legobot, Kilom691, AnomieBOT, Erel Segal, Citation bot, SteveWoolf, Undsoweiter, FrescoBot, Nicolas Perrault III, Confluente, Ri-cardo Ferreira de Oliveira, Throw it in the Fire, Gnathan87, Setitup, EmausBot, John of Reading, Febuiles, Thecheesykid, ZéroBot,Chharvey, The man who was Friday, SporkBot, Zfeinst, Rathgemz, CocuBot, Vdamanafshan, Mesoderm, MerlIwBot, Wbm1058, Jak-shap, Paolo Lipparini, ElphiBot, Larion Garaczi, Aabhis, Jochen Burghardt, Mark viking, Eamonford, Sgbmyr, K401sTL3, Tudor987,Victor Lesyk, Some1Redirects4You and Anonymous: 79

• Preorder Source: https://en.wikipedia.org/wiki/Preorder?oldid=677053946 Contributors: AxelBoldt, Patrick, Repton, Delirium, An-dres, Dysprosia, Greenrd, Big Bob the Finder, BenRG, Tobias Bergemann, Giftlite, Markus Krötzsch, Lethe, Fropuff, Vadmium, De-fLog~enwiki, Zzo38, Jh51681, Barnaby dawson, Paul August, EmilJ, Msh210, Melaen, Joriki, Linas, Dionyziz, Mandarax, Salix alba,Cjoev, VKokielov, Mathbot, Jrtayloriv, YurikBot, Laurentius, Hairy Dude, WikidSmaht, Trovatore, Modify, Netrapt, Wasseralm, Smack-Bot, XudongGuan~enwiki, DCary, Jdthood, Mets501, PaulGS, Stotr~enwiki, Zero sharp, CRGreathouse, Michael A. White, Magioladitis,David Eppstein, Jwuthe2, PaulTanenbaum, SieBot, Thehotelambush, MenoBot, Functor salad, He7d3r, Sun Creator, Cenarium, 1ForThe-Money, Palnot, Мыша, Legobot, Luckas-bot, AnomieBOT, DannyAsher, Xqbot, VladimirReshetnikov, ComputScientist, BrideOfKrip-kenstein, Notedgrant, WikitanvirBot, Lclem, Dfabera, SporkBot, Paolo Lipparini, RichardMills65, Khazar2, Lerutit, Jochen Burghardt,Reatlas, Damonamc and Anonymous: 26

• Prewellordering Source: https://en.wikipedia.org/wiki/Prewellordering?oldid=488790287Contributors: Zundark, Patrick, Charles Matthews,EmilJ, Oleg Alexandrov, NickBush24, Trovatore, Tetracube, That Guy, From That Show!, SmackBot, Nbarth, Henry Delforn (old), Pal-not, Citation bot and Anonymous: 3

• Process calculus Source: https://en.wikipedia.org/wiki/Process_calculus?oldid=653588725Contributors: Michael Hardy, AlexR, Theresaknott, Charles Matthews, Zoicon5, Phil Boswell, Wmahan, Neilc, Solitude, Leibniz, Linas, Ruud Koot, MarSch, XP1, Jameshfisher,Vonkje, GangofOne, Wavelength, Koffieyahoo, CarlHewitt, Gareth Jones, RabidDeity, Jpbowen, SockPuppetVandal, Voidxor, Misza13,SmackBot, Chris the speller, Nbarth, Allan McInnes, Ezrakilty, Sam Staton, Blaisorblade, Thijs!bot, Dougher, Skraedingie, Barkjon,Roxy the dog, MystBot, Addbot, Tassedethe, Lightbot, Yobot, AnomieBOT, GrouchoBot, BehnazCh, Vasywriter, Iæfai, WikitanvirBot,Clayrat, Serketan, Bethnim, Helpful Pixie Bot, Ulidtko and Anonymous: 39

• Propositional function Source: https://en.wikipedia.org/wiki/Propositional_function?oldid=668907292 Contributors: Michael Hardy,TakuyaMurata, Jason Quinn, Rgdboer, Linas, Thekohser, Arthur Rubin, Gregbard, Uncle uncle uncle, Izno, Addbot, Montblanc1988,Orenburg1, Helpful Pixie Bot, YFdyh-bot, JYBot and Anonymous: 3

https://en.wikipedia.org/wiki/Inverse_trigonometric_functions?oldid=676298808

https://en.wikipedia.org/wiki/Near_sets?oldid=662493389

https://en.wikipedia.org/wiki/Partial_equivalence_relation?oldid=672263741

https://en.wikipedia.org/wiki/Partial_function?oldid=670578722

https://en.wikipedia.org/wiki/Partially_ordered_set?oldid=672760099

https://en.wikipedia.org/wiki/Preorder?oldid=677053946

https://en.wikipedia.org/wiki/Prewellordering?oldid=488790287

https://en.wikipedia.org/wiki/Process_calculus?oldid=653588725

https://en.wikipedia.org/wiki/Propositional_function?oldid=668907292


• Quasi-commutative property Source: https://en.wikipedia.org/wiki/Quasi-commutative_property?oldid=643466557Contributors: MichaelHardy, Dirac66, Marius siuram and OccultZone

• Quasitransitive relation Source: https://en.wikipedia.org/wiki/Quasitransitive_relation?oldid=625289154Contributors: Patrick, CharlesMatthews, Giftlite, Btyner, Salix alba, SmackBot, Zahid Abdassabur, CRGreathouse, CBM, Gregbard, Sam Staton, Erik9bot, Deltahe-dron, Jochen Burghardt and Anonymous: 1

• Quotient by an equivalence relation Source: https://en.wikipedia.org/wiki/Quotient_by_an_equivalence_relation?oldid=657098718Contributors: Michael Hardy, TakuyaMurata, Bearcat, D.Lazard, K9re11, Iaritmioawp and Fimatic

• Rational consequence relation Source: https://en.wikipedia.org/wiki/Rational_consequence_relation?oldid=610309178 Contributors:Michael Hardy, Oleg Alexandrov, Reetep, SmackBot, CmdrObot, Gregbard, Enlil2, DavidCBryant, Carriearchdale and Tijfo098

• Reduct Source: https://en.wikipedia.org/wiki/Reduct?oldid=618205741 Contributors: John Baez, Spring Rubber, SmackBot, VaughanPratt, Tikiwont, Hans Adler, Backslash Forwardslash, Gf uip, EmausBot, Bgeron, Monkbot and Anonymous: 3

• Reflexive closure Source: https://en.wikipedia.org/wiki/Reflexive_closure?oldid=627004515 Contributors: Timwi, Arthur Rubin, Girl-withglasses, Henry Delforn (old), Addbot, Dawynn, Pcap, Renato sr, GrouchoBot, EmausBot, Jochen Burghardt, Vaizar and Anonymous:2

• Reflexive relation Source: https://en.wikipedia.org/wiki/Reflexive_relation?oldid=675919688Contributors: AxelBoldt, DavidSJ, Patrick,Wshun, TakuyaMurata, Looxix~enwiki, William M. Connolley, Charles Matthews, Josh Cherry, MathMartin, Henrygb, Tobias Berge-mann, Giftlite, Jason Quinn, Gubbubu, Urhixidur, Ascánder, Paul August, BenjBot, Spayrard, Spoon!, Jet57, LavosBacons, Wtmitchell,Bookandcoffee, Oleg Alexandrov, Joriki, Mel Etitis, LOL, MFH, Isnow, Audiovideo, Margosbot~enwiki, Fresheneesz, Chobot, Yurik-Bot, Laurentius, Maelin, Mathlaura, KarlHeg, Arthur Rubin, Isaac Dupree, Jdthood, Mhym, Ceosion, Mike Fikes, Fjbex, CRGreathouse,CBM, Gregbard, Farzaneh, Wikid77, JAnDbot, Policron, Joshua Issac, VolkovBot, Jackfork, Jamelan, Ocsenave, SieBot, Davidellerman,Henry Delforn (old), Hello71, Cuyaken, ClueBot, Ywanne, Da rulz07, SoxBot III, WikHead, Addbot, Download, Favonian, Luckas-bot, Yobot, Renato sr, Pkukiss, Galoubet, ArthurBot, Xqbot, Z0973, I dream of horses, RedBot, MastiBot, Gamewizard71, EmausBot,Dmayank, DimitriC, ClueBot NG, Kasirbot, Joel B. Lewis, BG19bot, Solomon7968, ChrisGualtieri, Eptified, Lerutit, Jochen Burghardt,Seanhalle and Anonymous: 37

• Relation algebra Source: https://en.wikipedia.org/wiki/Relation_algebra?oldid=657291951Contributors: Zundark, Michael Hardy, AugPi,Charles Matthews, Tobias Bergemann, Lethe, Mboverload, D6, Elwikipedista~enwiki, Giraffedata, AshtonBenson, Woohookitty, PaulCarpenter, BD2412, Rjwilmsi, Koavf, Tillmo, Ott2, Mhss, Concerned cynic, Nbarth, Jon Awbrey, Lambiam, Physis, Mets501, VaughanPratt, CBM, Gregbard, Sam Staton, King Bee, JustAGal, Balder ten Cate, David Eppstein, R'n'B, Leyo, Ramsey2006, Plasticup, John-Blackburne, The Tetrast, Linelor, Hans Adler, Addbot, QuadrivialMind, Yobot, AnomieBOT, Nastor, LilHelpa, Xqbot, Samppi111,Charvest, FrescoBot, Irmy, Sjcjoosten, SchreyP, Brad7777, Khazar2, Lerutit, RPI, JaconaFrere, SaltHerring, Some1Redirects4You andAnonymous: 35

• Relation construction Source: https://en.wikipedia.org/wiki/Relation_construction?oldid=564892457 Contributors: Charles Matthews,Carlossuarez46, Kaldari, Paul August, El C, DoubleBlue, Nihiltres, TeaDrinker, Gwernol, Wknight94, Closedmouth, SmackBot, JonAwbrey, JzG, Coredesat, Slakr, CBM, Gogo Dodo, Hut 8.5, Brusegadi, David Eppstein, Brigit Zilwaukee, Yolanda Zilwaukee, Ars Tottle,The Proposition That, Spellcast, Corvus cornix, Seb26, Maelgwnbot, Blanchardb, RABBU, REBBU, DEBBU, DABBUØ, Wolf of theSteppes, REBBUØ, Doubtentry, Education Is The Basis Of Law And Order, Bare In Mind, Preveiling Opinion Of Dominant OpinionGroup, VOCØ, Buchanan’s Navy Sec, Kaiba, Marsboat, Trainshift, Pluto Car, Unco Guid, Viva La Information Revolution!, FlowerMound Belle, Editortothemasses, Navy Pierre, Mrs. Lovett’s Meat Puppets, Unknown Justin, West Goshen Guy, Southeast Penna Poppa,Delaware Valley Girl, Erik9bot and Lerutit

• Representation (mathematics) Source: https://en.wikipedia.org/wiki/Representation_(mathematics)?oldid=647042099 Contributors:Michael Hardy, Giftlite, El C, Linas, SixWingedSeraph, Rjwilmsi, MarSch, Reyk, Magioladitis, A3nm, David Eppstein, PaulTanenbaum,Geometry guy, SieBot, Addbot, Twri, Trappist the monk, Helpful Pixie Bot, BattyBot, Mrt3366 and Anonymous: 2

• Semiorder Source: https://en.wikipedia.org/wiki/Semiorder?oldid=607501955 Contributors: Rjwilmsi, Headbomb, David Eppstein,Undsoweiter, Joel B. Lewis, Lerutit and Anonymous: 2

• Separoid Source: https://en.wikipedia.org/wiki/Separoid?oldid=588579933 Contributors: Michael Hardy, Salix alba, SmackBot, Mhym,Gregbard, Magioladitis, Qworty, Hans Adler and Strausz~enwiki

• Sequential composition Source: https://en.wikipedia.org/wiki/Process_calculus?oldid=653588725Contributors: Michael Hardy, AlexR,Theresa knott, Charles Matthews, Zoicon5, Phil Boswell, Wmahan, Neilc, Solitude, Leibniz, Linas, Ruud Koot, MarSch, XP1, Jamesh-fisher, Vonkje, GangofOne, Wavelength, Koffieyahoo, CarlHewitt, Gareth Jones, RabidDeity, Jpbowen, SockPuppetVandal, Voidxor,Misza13, SmackBot, Chris the speller, Nbarth, Allan McInnes, Ezrakilty, Sam Staton, Blaisorblade, Thijs!bot, Dougher, Skraedingie,Barkjon, Roxy the dog, MystBot, Addbot, Tassedethe, Lightbot, Yobot, AnomieBOT, GrouchoBot, BehnazCh, Vasywriter, Iæfai, Wiki-tanvirBot, Clayrat, Serketan, Bethnim, Helpful Pixie Bot, Ulidtko and Anonymous: 39

• Series-parallel partial order Source: https://en.wikipedia.org/wiki/Series-parallel_partial_order?oldid=629384776 Contributors: Za-slav, A3nm, David Eppstein, Trappist the monk, Helpful Pixie Bot, Deltahedron and Anonymous: 1

• Surjective function Source: https://en.wikipedia.org/wiki/Surjective_function?oldid=672639860Contributors: AxelBoldt, Tarquin, Amil-lar, XJaM, Toby Bartels, Michael Hardy, Wshun, Pit~enwiki, Karada, Александър, Glenn, Jeandré du Toit, Hashar, Hawthorn, CharlesMatthews, Dysprosia, David Shay, Ed g2s, Phil Boswell, Aleph4, Robbot, Fredrik, Tobias Bergemann, Giftlite, Lethe, Jason Quinn,Jorge Stolfi, Matt Crypto, Keeyu, Rheun, MarkSweep, AmarChandra, Tsemii, TheObtuseAngleOfDoom, Vivacissamamente, Rich Farm-brough, Quistnix, Paul August, Bender235, Nandhp, Kevin Lamoreau, Larry V, Obradovic Goran, Dallashan~enwiki, ABCD, Schapel,Oleg Alexandrov, Tbsmith, Mindmatrix, LOL, Rjwilmsi, MarSch, FlaBot, Chobot, Manscher, Algebraist, Angus Lepper, Ksnortum,Rick Norwood, Sbyrnes321, SmackBot, Rotemliss, Bluebot, Javalenok, TedE, Soapergem, Dreadstar, Saippuakauppias, MickPurcell,16@r, Inquisitus, CBM, MatthewMain, Gregbard, Marqueed, Sam Staton, Pjvpjv, Prolog, Salgueiro~enwiki, JAnDbot, JamesBWatson,JJ Harrison, Martynas Patasius, MartinBot, TechnoFaye, Malerin, Dubhe.sk, Theabsurd, UnicornTapestry, Eliuha gmail.com, AnonymousDissident, SieBot, SLMarcus, Paolo.dL, Peiresc~enwiki, Classicalecon, UKoch, Watchduck, Bender2k14, SchreiberBike, Neuralwarp,Petru Dimitriu, Matthieumarechal, Kal-El-Bot, Addbot, Download, PV=nRT, ,ماني Zorrobot, Jarble, Legobot, Luckas-bot, Yobot, Frag-gle81, II MusLiM HyBRiD II, Xqbot, TechBot, Shvahabi, Raffamaiden, Omnipaedista, Applebringer, Erik9bot, LucienBOT, Tbhotch,Xnn, Jowa fan, EmausBot, PrisonerOfIce, WikitanvirBot, GoingBatty, Sasuketiimer, Maschen, Mjbmrbot, Anita5192, ClueBot NG,Helpful Pixie Bot, BG19bot, Cispyre, Lfahlberg, JPaestpreornJeolhlna, TranquilHope and Anonymous: 87

https://en.wikipedia.org/wiki/Quasi-commutative_property?oldid=643466557

https://en.wikipedia.org/wiki/Quasitransitive_relation?oldid=625289154

https://en.wikipedia.org/wiki/Quotient_by_an_equivalence_relation?oldid=657098718

https://en.wikipedia.org/wiki/Rational_consequence_relation?oldid=610309178

https://en.wikipedia.org/wiki/Reduct?oldid=618205741

https://en.wikipedia.org/wiki/Reflexive_closure?oldid=627004515

https://en.wikipedia.org/wiki/Reflexive_relation?oldid=675919688

https://en.wikipedia.org/wiki/Relation_algebra?oldid=657291951

https://en.wikipedia.org/wiki/Relation_construction?oldid=564892457

https://en.wikipedia.org/wiki/Representation_(mathematics)?oldid=647042099

https://en.wikipedia.org/wiki/Semiorder?oldid=607501955

https://en.wikipedia.org/wiki/Separoid?oldid=588579933

https://en.wikipedia.org/wiki/Process_calculus?oldid=653588725

https://en.wikipedia.org/wiki/Series-parallel_partial_order?oldid=629384776

https://en.wikipedia.org/wiki/Surjective_function?oldid=672639860


• Symmetric closure Source: https://en.wikipedia.org/wiki/Symmetric_closure?oldid=627004592 Contributors: Michael Hardy, Timwi,Addbot, Pcap, ZéroBot, YFdyh-bot, Jochen Burghardt and Anonymous: 2

• Symmetric relation Source: https://en.wikipedia.org/wiki/Symmetric_relation?oldid=676529262 Contributors: Patrick, Looxix~enwiki,William M. Connolley, Charles Matthews, MathMartin, Tobias Bergemann, Giftlite, Elektron, Ascánder, Paul August, Syp, Joriki, Is-now, Salix alba, Margosbot~enwiki, Fresheneesz, Roboto de Ajvol, Laurentius, Bota47, Arthur Rubin, Incnis Mrsi, Unyoyega, Jdthood,Vina-iwbot~enwiki, Gregbard, Thijs!bot, Mouchoir le Souris, David Eppstein, Jamelan, Henry Delforn (old), ClueBot, Libcub, Addbot,Luckas-bot, ArthurBot, Xqbot, Adavis444, RedBot, EmausBot, ZéroBot, Zap Rowsdower, EdoBot, DASHBotAV, 28bot, Kasirbot, No-suchforever, Aryan5496 and Anonymous: 18

• Ternary equivalence relation Source: https://en.wikipedia.org/wiki/Ternary_equivalence_relation?oldid=672070603Contributors: Mel-choir

• Ternary relation Source: https://en.wikipedia.org/wiki/Ternary_relation?oldid=677556030Contributors: Michael Hardy, Charles Matthews,Tobias Bergemann, Ancheta Wis, Abdull, Paul August, El C, Rgdboer, Versageek, Oleg Alexandrov, Jeffrey O. Gustafson, RxS, Dou-bleBlue, TeaDrinker, Wknight94, Closedmouth, Luk, Sardanaphalus, SmackBot, KnowledgeOfSelf, Melchoir, C.Fred, BiT, Aksi great,Nbarth, Jon Awbrey, Lambiam, JzG, Tim Q. Wells, Slakr, Politepunk, General Eisenhower, Happy-melon, Tawkerbot2, CBM, GogoDodo, Alaibot, Luna Santin, Hut 8.5, Transcendence, Brusegadi, David Eppstein, JoergenB, Santiago Saint James, Brigit Zilwaukee,Yolanda Zilwaukee, Fallopius Manque, Mike V, CardinalDan, Rei-bot, Seb26, GlobeGores, Lucien Odette, REBBU, RABBUØ, Wolfof the Steppes, REBBUØ, Doubtentry, DEBBUØ, Education Is The Basis Of Law And Order, Bare In Mind, Preveiling Opinion OfDominant Opinion Group, Hans Adler, Buchanan’s Navy Sec, Mr. Peabody’s Boy, Overstay, Marsboat, Unco Guid, Viva La InformationRevolution!, Autocratic Uzbek, Poke Salat Annie, Flower Mound Belle, Navy Pierre, Mrs. Lovett’s Meat Puppets, Unknown Justin, IPPhreely, West Goshen Guy, Delaware Valley Girl, Addbot, Jarble, Yobot, Vini 17bot5, AnomieBOT, In digma, Erik9bot, AManWithNo-Plan and Anonymous: 5

• Tolerance relation Source: https://en.wikipedia.org/wiki/Tolerance_relation?oldid=662333091 Contributors: Michael Hardy, Rjwilmsi,SmackBot, NickPenguin, Floridi~enwiki, Classicalecon, Gabrno, Clone200 and Anonymous: 1

• Total order Source: https://en.wikipedia.org/wiki/Total_order?oldid=673384381 Contributors: Damian Yerrick, AxelBoldt, Zundark,XJaM, Fritzlein, Patrick, Michael Hardy, Dori, AugPi, Dysprosia, Jitse Niesen, Greenrd, Zoicon5, Hyacinth, VeryVerily, Fibonacci,McKay, Aleph4, Gandalf61, MathMartin, Rursus, Tobias Bergemann, Giftlite, Mshonle~enwiki, Markus Krötzsch, Lethe, Waltpohl,DefLog~enwiki, Alberto da Calvairate~enwiki, Quarl, Elroch, Paul August, Susvolans, Army1987, Func, Cmdrjameson, Msh210, Pion,Joriki, MattGiuca, Yurik, OneWeirdDude, Salix alba, VKokielov, Mathbot, Margosbot~enwiki, Wastingmytime, Chobot, YurikBot,Hede2000, Tetracube, Rdore, Melchoir, Gelingvistoj, Mhss, Chris the speller, Bazonka, Jdthood, Javalenok, Michael Kinyon, Loadmas-ter, Mets501, JRSpriggs, George100, CRGreathouse, CBM, Thomasmeeks, Oryanw~enwiki, VectorPosse, JAnDbot, David Eppstein,Infovarius, Osquar F, PaulTanenbaum, SieBot, Ceroklis, Anchor Link Bot, Heinzi.at, WurmWoode, Universityuser, Palnot, Marc vanLeeuwen, Addbot, Tanhabot, AsphyxiateDrake, Luckas-bot, Yobot, Charlatino, White gecko, 1exec1, Infvwl, GrouchoBot, Jsjunkie,Quondum, D.Lazard, SporkBot, CocuBot, BG19bot, YumOooze, YFdyh-bot, Austinfeller, Mark viking, नितीश् चन्द्र and Anonymous:49

• Total relation Source: https://en.wikipedia.org/wiki/Total_relation?oldid=608974105 Contributors: Patrick, Charles Matthews, Dcoet-zee, Jitse Niesen, Tobias Bergemann, Lethe, Alberto da Calvairate~enwiki, Paul August, Ntmatter, Oleg Alexandrov, Joriki, Salix alba,Nneonneo, Mathbot, Fresheneesz, Bota47, Jdthood, Stotr~enwiki, JAnDbot, TXiKiBoT, Jamelan, Hans Adler, Erodium, Addbot, Fres-coBot, SporkBot, Helpful Pixie Bot, Deltahedron, Kephir and Anonymous: 11

• Transitive closure Source: https://en.wikipedia.org/wiki/Transitive_closure?oldid=675927373 Contributors: Awaterl, Vkuncak, Patrick,Michael Hardy, Charles Matthews, Timwi, Dcoetzee, Populus, Borislav, Tobias Bergemann, Giftlite, Fropuff, Matt Crypto, Alexf,Quickwik, Creidieki, Obradovic Goran, Oleg Alexandrov, Joriki, Neonfreon, Salix alba, AL SAM, Bgwhite, Ott2, Arthur Rubin, Plas-ticphilosopher, KnightRider~enwiki, Mhss, Plustgarten, Dreadstar, NeilFraser, Lyonsam, Loadmaster, JRSpriggs, CRGreathouse, CBM,ShelfSkewed, Gregbard, Girlwithglasses, Kirtag Hratiba, Thijs!bot, JAnDbot, A3nm, David Eppstein, Yavoh, Cometstyles, VolkovBot,Sdrucker, PaulTanenbaum, Jamelan, Tomaxer, LungZeno, Henry Delforn (old), DuaneLAnderson, CBM2, Classicalecon, ClueBot, Ben-der2k14, PixelBot, AmirOnWiki, MountainGoat8, Tayste, Addbot, Luckas-bot, AnomieBOT, BenzolBot, RedBot, MastiBot, Trappist themonk, Wizeguytristram, Quondum, Tijfo098, ClueBot NG, BG19bot, Solomon7968, Danwizard208, Dmitri L. Slabk., Vpieterse~enwiki,Seahen, Artdadamo and Anonymous: 26

• Transitive relation Source: https://en.wikipedia.org/wiki/Transitive_relation?oldid=677849508Contributors: Zundark, Patrick, MichaelHardy, Rp, Looxix~enwiki, Andres, Charles Matthews, Dcoetzee, Jitse Niesen, Fredrik, MathMartin, Tobias Bergemann, Giftlite, Ben-FrantzDale, Gubbubu, Chowbok, Paul August, MyNameIsNotBob, Spoon!, Polluks, DanShearer, Woohookitty, Linas, LOL, Isnow,Palica, Jérémie Lumbroso~enwiki, Salix alba, Mathbot, Fresheneesz, YurikBot, Laurentius, Sasuke Sarutobi, 48v, Bota47, Arthur Rubin,MaratL, Wasseralm, JJL, SmackBot, InverseHypercube, Nbarth, Wen D House, Cybercobra, Jóna Þórunn, Lambiam, Coredesat, Lyon-sam, Cbuckley, CRGreathouse, Aggarwal kshitij, CBM, Thomasmeeks, Gogo Dodo, Tawkerbot4, AntiVandalBot, Mhaitham.shammaa,MER-C, .anacondabot, Magioladitis, Albmont, David Eppstein, Edward321, MartinBot, Extransit, Tomaz.slivnik, Policron, VolkovBot,AThomas203, Jamelan, Cnilep, SieBot, Paradoctor, Henry Delforn (old), Anchor Link Bot, ClueBot, Tomvanderweide, Sarbogard, Ot-tawahitech, Alexbot, Wikibojopayne, Pa68, SilvonenBot, Addbot, Luckas-bot, Yobot, Ptbotgourou, Materialscientist, GrouchoBot, Und-soweiter, RedBot, Katovatzschyn, EmausBot, Slightsmile, IGeMiNix, ChuispastonBot, ClueBot NG, Pars99, Sourabh.khot, Justincheng12345-bot, Lerutit, Loraof and Anonymous: 61

• Trichotomy (mathematics) Source: https://en.wikipedia.org/wiki/Trichotomy_(mathematics)?oldid=646849767Contributors: Zundark,Patrick, Michael Hardy, Arthur Frayn, Casu Marzu, Henrygb, UtherSRG, Tobias Bergemann, Macrakis, Paul August, Oleg Alexandrov,VKokielov, Michael Slone, Bota47, JJL, SmackBot, Mhss, Colonies Chris, Jdthood, Cybercobra, Courcelles, JRSpriggs, Meng.benjamin,Gregbard, Difluoroethene, David Cherney, AntiVandalBot, Indeed123, Minnnnng, YohanN7, MilesAgain, Addbot, Luckas-bot, AnomieBOT,Götz, Xqbot, Constructive editor, ComputScientist, Þjóðólfr, Tkuvho, SporkBot, Paulmiko, Helpful Pixie Bot, Zorglub x and Anonymous:23

• Unimodality Source: https://en.wikipedia.org/wiki/Unimodality?oldid=670019894Contributors: Michael Hardy, Komap, Henrygb, Com-mander Keane, Rjwilmsi, Bhny, Tropylium, Wen D House, Henning Makholm, JzG, CRGreathouse, Magioladitis, David Eppstein, DrMi-cro, Asaduzaman, Melcombe, Muhandes, Juanm55, Addbot, Nate Wessel, Yobot, Flavonoid, Isheden, FrescoBot, Ofir michael, Duoduo-duo, Gypped, Tgoossens, Helpful Pixie Bot, BG19bot, Op47, CitationCleanerBot, BattyBot, ChrisGualtieri, Iamyatin and Anonymous:14

https://en.wikipedia.org/wiki/Symmetric_closure?oldid=627004592

https://en.wikipedia.org/wiki/Symmetric_relation?oldid=676529262

https://en.wikipedia.org/wiki/Ternary_equivalence_relation?oldid=672070603

https://en.wikipedia.org/wiki/Ternary_relation?oldid=677556030

https://en.wikipedia.org/wiki/Tolerance_relation?oldid=662333091

https://en.wikipedia.org/wiki/Total_order?oldid=673384381

https://en.wikipedia.org/wiki/Total_relation?oldid=608974105

https://en.wikipedia.org/wiki/Transitive_closure?oldid=675927373

https://en.wikipedia.org/wiki/Transitive_relation?oldid=677849508

https://en.wikipedia.org/wiki/Trichotomy_(mathematics)?oldid=646849767

https://en.wikipedia.org/wiki/Unimodality?oldid=670019894


• Weak ordering Source: https://en.wikipedia.org/wiki/Weak_ordering?oldid=640088882 Contributors: Patrick, Michael Hardy, Chinju,Dcoetzee, MathMartin, Tobias Bergemann, Pretzelpaws, AlphaEtaPi, Zaslav, Aisaac, YurikBot, Gadget850, Modify, Oli Filth, Jdthood,Chlewbot, Radiant chains, Jafet, CRGreathouse, Sdorrance, Gregbard, Widefox, Medinoc, Zeitlupe, David Eppstein, Jonathanrcoxhead,Watchduck, Addbot, Kne1p, Forich, Citation bot, ArthurBot, Howard McCay, Citation bot 1, SporkBot, Joel B. Lewis, Helpful Pixie Bot,Jochen Burghardt, JustBerry and Anonymous: 12

• Well-founded relation Source: https://en.wikipedia.org/wiki/Well-founded_relation?oldid=659840399 Contributors: The Anome, Ap,Michael Hardy, Dominus, TakuyaMurata, Cyp, Charles Matthews, VeryVerily, Aleph4, Mountain, Tobias Bergemann, Filemon, Marekpetrik,Lethe, Lupin, Waltpohl, Mani1, Paul August, EmilJ, Nahabedere, MZMcBride, R.e.b., FlaBot, Trovatore, Mikeblas, Crasshopper, Mar-lasdad, That Guy, From That Show!, SmackBot, Ron.garcia, Nbarth, Mmehdi.g, Mets501, CBM, WillowW, Pgagge, JAnDbot, Albmont,Leyo, Reedy Bot, Alexsmail, Thehotelambush, Mutilin, Addbot, KamikazeBot, RibotBOT, FrescoBot, Involutive-revolution, MerlIwBot,Wohlfundi, YiFeiBot, Some1Redirects4You and Anonymous: 22

• Well-order Source: https://en.wikipedia.org/wiki/Well-order?oldid=672639066 Contributors: AxelBoldt, Mav, Zundark, Josh Grosse,Patrick, Michael Hardy, David Martland, Dominus, TakuyaMurata, Andres, Vargenau, Revolver, Charles Matthews, Timwi, Populus,Aleph4, R3m0t, MathMartin, Tobias Bergemann, Tosha, Giftlite, Dbenbenn, Ian Maxwell, Lethe, Arturus~enwiki, Jorend, Karl-Henner,Rich Farmbrough, Luqui, Paul August, Sligocki, RJFJR, Eyu100, Salix alba, FlaBot, Margosbot~enwiki, Chobot, YurikBot, Hairy Dude,Trovatore, Obey, Bota47, Arthur Rubin, MullerHolk, Ghazer~enwiki, GrinBot~enwiki, KnightRider~enwiki, Alan McBeth, Gelingvistoj,Mhss, Nbarth, Loodog, Jim.belk, Loadmaster, JRSpriggs, CBM, WeggeBot, Myasuda, Thijs!bot, Nadav1, Escarbot, Albmont, Odexios,VolkovBot, Don4of4, SieBot, Rumping, Fyyer, Bender2k14, His Wikiness, Palnot, Addbot, Luckas-bot, Yobot, Ptbotgourou, Jarmiz,Xqbot, GrouchoBot, Miyagawa, Adrionwells, RjwilmsiBot, Honestrosewater, Hunterbd, SporkBot, Misshamid, ChrisGualtieri, Khazar2,Jose Brox and Anonymous: 29

• Well-quasi-ordering Source: https://en.wikipedia.org/wiki/Well-quasi-ordering?oldid=677173775Contributors: Patrick, Chinju, CharlesMatthews, Tobias Bergemann, Peter Kwok, Rich Farmbrough, Paul August, EmilJ, R.e.b., Open2universe, PhS, That Guy, From ThatShow!, Mets501, Pierre de Lyon, David Eppstein, Kope, R'n'B, Alexwright, Fcarreiro, Niceguyedc, Palnot, Addbot, DOI bot, Citationbot, FrescoBot, Citation bot 1, Gongfarmerzed, John of Reading, ZéroBot, Ɯ, Mastergreg82, Paolo Lipparini, CitationCleanerBot, JeffErickson, Mark viking, Anrnusna, ,תמשל Gasarch and Anonymous: 5

78.10.2 Images• File:13-Weak-Orders.svg Source: https://upload.wikimedia.org/wikipedia/commons/3/3c/13-Weak-Orders.svg License: Public do-

main Contributors: Transferred from en.wikipedia to Commons. Original artist: SVG version created by Jafet.vixle at en.wikipedia,based on a public-domain PNG version created 2006-0 9-14 by David Eppstein

• File:Ambox_important.svg Source: https://upload.wikimedia.org/wikipedia/commons/b/b4/Ambox_important.svg License: Public do-main Contributors: Own work, based off of Image:Ambox scales.svg Original artist: Dsmurat (talk · contribs)

• File:Arcsecant_Arccosecant.svg Source: https://upload.wikimedia.org/wikipedia/commons/5/56/Arcsecant_Arccosecant.svg License:CC BY-SA 3.0 Contributors: Own work Original artist: Geek3

• File:Arcsine_Arccosine.svg Source: https://upload.wikimedia.org/wikipedia/commons/b/b4/Arcsine_Arccosine.svg License: CC BY-SA 3.0 Contributors: Own work Original artist: Geek3

• File:Arctangent_Arccotangent.svg Source: https://upload.wikimedia.org/wikipedia/commons/9/9a/Arctangent_Arccotangent.svg Li-cense: CC BY-SA 3.0 Contributors: Own work Original artist: Geek3

• File:Bijection.svg Source: https://upload.wikimedia.org/wikipedia/commons/a/a5/Bijection.svg License: Public domain Contributors:enwiki Original artist: en:User:Schapel

• File:Bijective_composition.svg Source: https://upload.wikimedia.org/wikipedia/commons/a/a2/Bijective_composition.svgLicense: Pub-lic domain Contributors: ? Original artist: ?

• File:Bimodal.png Source: https://upload.wikimedia.org/wikipedia/commons/e/e2/Bimodal.png License: CC-BY-SA-3.0 Contributors:? Original artist: ?

• File:Bimodal_geological.PNG Source: https://upload.wikimedia.org/wikipedia/commons/b/bc/Bimodal_geological.PNGLicense: Pub-lic domain Contributors: Own work Original artist: Tungsten

• File:Birkhoff120.svg Source: https://upload.wikimedia.org/wikipedia/commons/7/7c/Birkhoff120.svg License: Public domain Contrib-utors: Own work Original artist: David Eppstein

• File:Bothodd.png Source: https://upload.wikimedia.org/wikipedia/en/d/de/Bothodd.png License: PD Contributors:Own workOriginal artist:Fresheneesz (talk) (Uploads)

• File:CardContin.svg Source: https://upload.wikimedia.org/wikipedia/commons/7/75/CardContin.svg License: Public domain Contrib-utors: en:Image:CardContin.png Original artist: en:User:Trovatore, recreated by User:Stannered

• File:Cercle_trigo.png Source: https://upload.wikimedia.org/wikipedia/commons/0/05/Cercle_trigo.png License: CC-BY-SA-3.0 Con-tributors: réalisé avec un programme de dessin vectoriel par Cdang Original artist: Christophe Dang Ngoc Chan Cdang at fr.wikipedia

• File:Codomain2.SVG Source: https://upload.wikimedia.org/wikipedia/commons/6/64/Codomain2.SVG License: Public domain Con-tributors: Transferred from en.wikipediaOriginal artist: Damien Karras (talk). Original uploader was Damien Karras at en.wikipedia

• File:Commons-logo.svg Source: https://upload.wikimedia.org/wikipedia/en/4/4a/Commons-logo.svg License: ? Contributors: ? Origi-nal artist: ?

• File:Commutative_Addition.svg Source: https://upload.wikimedia.org/wikipedia/commons/3/36/Commutative_Addition.svg License:CC-BY-SA-3.0 Contributors: self-made using previous GFDL work by Melchoir Original artist: Weston.pace. Attribution: Apples inimage were created by Melchoir

https://en.wikipedia.org/wiki/Weak_ordering?oldid=640088882

https://en.wikipedia.org/wiki/Well-founded_relation?oldid=659840399

https://en.wikipedia.org/wiki/Well-order?oldid=672639066

https://en.wikipedia.org/wiki/Well-quasi-ordering?oldid=677173775

https://upload.wikimedia.org/wikipedia/commons/3/3c/13-Weak-Orders.svg

http://en.wikipedia.org/

//en.wikipedia.org/wiki/User:Jafet.vixle


//en.wikipedia.org/wiki/Image:13-Weak-Orders.png

//commons.wikimedia.org/wiki/User:David_Eppstein

https://upload.wikimedia.org/wikipedia/commons/b/b4/Ambox_important.svg

//commons.wikimedia.org/wiki/File:Ambox_scales.svg

//commons.wikimedia.org/wiki/User:Dsmurat

//commons.wikimedia.org/wiki/User_talk:Dsmurat

//commons.wikimedia.org/wiki/Special:Contributions/Dsmurat

https://upload.wikimedia.org/wikipedia/commons/5/56/Arcsecant_Arccosecant.svg

//commons.wikimedia.org/wiki/User:Geek3

https://upload.wikimedia.org/wikipedia/commons/b/b4/Arcsine_Arccosine.svg


https://upload.wikimedia.org/wikipedia/commons/9/9a/Arctangent_Arccotangent.svg


https://upload.wikimedia.org/wikipedia/commons/a/a5/Bijection.svg

//en.wikipedia.org/wiki/User:Schapel

https://upload.wikimedia.org/wikipedia/commons/a/a2/Bijective_composition.svg

https://upload.wikimedia.org/wikipedia/commons/e/e2/Bimodal.png

https://upload.wikimedia.org/wikipedia/commons/b/bc/Bimodal_geological.PNG

//commons.wikimedia.org/w/index.php?title=User:Tungsten&action=edit&redlink=1

https://upload.wikimedia.org/wikipedia/commons/7/7c/Birkhoff120.svg


https://upload.wikimedia.org/wikipedia/en/d/de/Bothodd.png

//en.wikipedia.org/wiki/User:Fresheneesz

//en.wikipedia.org/wiki/User_talk:Fresheneesz

//en.wikipedia.org/wiki/Special:ListFiles/Fresheneesz

https://upload.wikimedia.org/wikipedia/commons/7/75/CardContin.svg

//en.wikipedia.org/wiki/Image:CardContin.png

//en.wikipedia.org/wiki/User:Trovatore

//commons.wikimedia.org/wiki/User:Stannered

https://upload.wikimedia.org/wikipedia/commons/0/05/Cercle_trigo.png

//fr.wikipedia.org/wiki/User:Cdang

//fr.wikipedia.org/wiki/User:Cdang

http://fr.wikipedia.org/

https://upload.wikimedia.org/wikipedia/commons/6/64/Codomain2.SVG


//en.wikipedia.org/wiki/User:Damien_Karras

//en.wikipedia.org/wiki/User_talk:Damien_Karras

//en.wikipedia.org/wiki/User:Damien_Karras


https://upload.wikimedia.org/wikipedia/en/4/4a/Commons-logo.svg

https://upload.wikimedia.org/wikipedia/commons/3/36/Commutative_Addition.svg

//commons.wikimedia.org/w/index.php?title=User:Weston.pace&action=edit&redlink=1


• File:Commutative_Word_Origin.PNG Source: https://upload.wikimedia.org/wikipedia/commons/d/da/Commutative_Word_Origin.PNG License: Public domain Contributors: Annales de Gergonne, Tome V, pg. 98 Original artist: Francois Servois

• File:Complex_ArcCot.jpg Source: https://upload.wikimedia.org/wikipedia/commons/6/60/Complex_ArcCot.jpg License: Public do-main Contributors: Eigenes Werk (own work) made with mathematica Original artist: Jan Homann

• File:Complex_ArcCsc.jpg Source: https://upload.wikimedia.org/wikipedia/commons/f/fd/Complex_ArcCsc.jpg License: Public do-main Contributors: Eigenes Werk (own work) made with mathematica Original artist: Jan Homann

• File:Complex_ArcSec.jpg Source: https://upload.wikimedia.org/wikipedia/commons/e/ec/Complex_ArcSec.jpg License: Public do-main Contributors: Eigenes Werk (own work) made with mathematica Original artist: Jan Homann

• File:Complex_arccos.jpg Source: https://upload.wikimedia.org/wikipedia/commons/4/4d/Complex_arccos.jpgLicense: Public domainContributors: made with mathematica, own work Original artist: Jan Homann

• File:Complex_arcsin.jpg Source: https://upload.wikimedia.org/wikipedia/commons/b/be/Complex_arcsin.jpg License: Public domainContributors: made with mathematica, own work Original artist: Jan Homann

• File:Complex_arctan.jpg Source: https://upload.wikimedia.org/wikipedia/commons/f/f5/Complex_arctan.jpg License: Public domainContributors: made with mathematica, own work Original artist: Jan Homann

• File:Compound_of_five_tetrahedra.png Source: https://upload.wikimedia.org/wikipedia/commons/6/6c/Compound_of_five_tetrahedra.png License: Attribution Contributors: Transferred from en.wikipedia. Original artist: ?

• File:Congruent_non-congruent_triangles.svg Source: https://upload.wikimedia.org/wikipedia/commons/1/12/Congruent_non-congruent_triangles.svg License: CC BY-SA 3.0 Contributors: Own work Original artist: Lfahlberg

• File:Descriptive.svg Source: https://upload.wikimedia.org/wikipedia/commons/c/c7/Descriptive.svg License: CC BY-SA 3.0 Contrib-utors: Own work Original artist: ChristopherJamesHenry

• File:Descriptively_Near_Sets.svg Source: https://upload.wikimedia.org/wikipedia/commons/8/8c/Descriptively_Near_Sets.svgLicense:CC BY-SA 3.0 Contributors: Own work Original artist: ChristopherJamesHenry

• File:Directed_set,_but_no_join_semi-lattice.png Source: https://upload.wikimedia.org/wikipedia/commons/a/a2/Directed_set%2C_but_no_join_semi-lattice.png License: CC BY-SA 3.0 Contributors: Own work Original artist: Jochen Burghardt

• File:Dynkin_Diagram_Isomorphisms.svg Source: https://upload.wikimedia.org/wikipedia/commons/0/03/Dynkin_Diagram_Isomorphisms.svg License: CC BY-SA 3.0 Contributors:

• Connected_Dynkin_Diagrams.svg Original artist: Connected_Dynkin_Diagrams.svg: R. A. Nonenmacher• File:E-to-the-i-pi.svg Source: https://upload.wikimedia.org/wikipedia/commons/3/35/E-to-the-i-pi.svg License: CC BY 2.5 Contribu-

tors: ? Original artist: ?• File:EANear_and_eAMer_Supercategories.png Source: https://upload.wikimedia.org/wikipedia/commons/f/f1/EANear_and_eAMer_

Supercategories.png License: CC BY-SA 3.0 Contributors: Own work Original artist: ChristopherJamesHenry• File:Edit-clear.svg Source: https://upload.wikimedia.org/wikipedia/en/f/f2/Edit-clear.svg License: Public domain Contributors: The

Tango! Desktop Project. Original artist:The people from the Tango! project. And according to the meta-data in the file, specifically: “Andreas Nilsson, and Jakub Steiner (althoughminimally).”

• File:Even_and_odd_antisymmetric_relation.png Source: https://upload.wikimedia.org/wikipedia/commons/c/cf/Even_and_odd_antisymmetric_relation.png License: Public domain Contributors: From the very same page I am now uploading it to. Original artist: Fresheneesz

• File:Folder_Hexagonal_Icon.svg Source: https://upload.wikimedia.org/wikipedia/en/4/48/Folder_Hexagonal_Icon.svg License: Cc-by-sa-3.0 Contributors: ? Original artist: ?

• File:Frigyes_Riesz.jpeg Source: https://upload.wikimedia.org/wikipedia/commons/c/cb/Frigyes_Riesz.jpeg License: Public domainContributors: ? Original artist: ?

• File:Graph_of_non-injective,_non-surjective_function_(red)_and_of_bijective_function_(green).gif Source: https://upload.wikimedia.org/wikipedia/commons/b/b0/Graph_of_non-injective%2C_non-surjective_function_%28red%29_and_of_bijective_function_%28green%29.gif License: CC BY-SA 3.0 Contributors: Own work Original artist: Jochen Burghardt

• File:GreaterThan.png Source: https://upload.wikimedia.org/wikipedia/en/5/59/GreaterThan.png License: Public domain Contributors:? Original artist: ?

• File:GreaterThanOrEqualTo.png Source: https://upload.wikimedia.org/wikipedia/en/0/0b/GreaterThanOrEqualTo.pngLicense: Pub-lic domain Contributors: ? Original artist: ?

• File:Hasse_diagram_of_powerset_of_3.svg Source: https://upload.wikimedia.org/wikipedia/commons/e/ea/Hasse_diagram_of_powerset_of_3.svg License: CC-BY-SA-3.0 Contributors: self-made using graphviz's dot. Original artist: KSmrq

• File:Hasse_diagram_of_powerset_of_3_no_greatest_or_least.svg Source: https://upload.wikimedia.org/wikipedia/commons/9/9e/Hasse_diagram_of_powerset_of_3_no_greatest_or_least.svg License: CC-BY-SA-3.0 Contributors: Based on File:Hasse diagram ofpowerset of 3.svg Original artist: User:Fibonacci, User:KSmrq

• File:Hypostasis-diagram.png Source: https://upload.wikimedia.org/wikipedia/commons/a/a2/Hypostasis-diagram.pngLicense: Publicdomain Contributors: Own work. Transferred from the English Wikipedia Original artist: User:Drini

• File:Infinite_lattice_of_divisors.svg Source: https://upload.wikimedia.org/wikipedia/commons/e/e6/Infinite_lattice_of_divisors.svgLicense: Public domain Contributors: w:de:Datei:Verband TeilerN.png Original artist: Watchduck (a.k.a. Tilman Piesk)

• File:Injection.svg Source: https://upload.wikimedia.org/wikipedia/commons/0/02/Injection.svg License: Public domain Contributors: ?Original artist: ?

• File:Injective_composition.svg Source: https://upload.wikimedia.org/wikipedia/commons/d/df/Injective_composition.svgLicense: Pub-lic domain Contributors: ? Original artist: ?

https://upload.wikimedia.org/wikipedia/commons/d/da/Commutative_Word_Origin.PNG

https://upload.wikimedia.org/wikipedia/commons/d/da/Commutative_Word_Origin.PNG

https://upload.wikimedia.org/wikipedia/commons/6/60/Complex_ArcCot.jpg

//commons.wikimedia.org/wiki/User:Jan_Homann

https://upload.wikimedia.org/wikipedia/commons/f/fd/Complex_ArcCsc.jpg


https://upload.wikimedia.org/wikipedia/commons/e/ec/Complex_ArcSec.jpg


https://upload.wikimedia.org/wikipedia/commons/4/4d/Complex_arccos.jpg

https://upload.wikimedia.org/wikipedia/commons/b/be/Complex_arcsin.jpg

https://upload.wikimedia.org/wikipedia/commons/f/f5/Complex_arctan.jpg

https://upload.wikimedia.org/wikipedia/commons/6/6c/Compound_of_five_tetrahedra.png

https://upload.wikimedia.org/wikipedia/commons/6/6c/Compound_of_five_tetrahedra.png


https://upload.wikimedia.org/wikipedia/commons/1/12/Congruent_non-congruent_triangles.svg

https://upload.wikimedia.org/wikipedia/commons/1/12/Congruent_non-congruent_triangles.svg

//commons.wikimedia.org/wiki/User:Lfahlberg

https://upload.wikimedia.org/wikipedia/commons/c/c7/Descriptive.svg

//commons.wikimedia.org/w/index.php?title=User:ChristopherJamesHenry&action=edit&redlink=1

https://upload.wikimedia.org/wikipedia/commons/8/8c/Descriptively_Near_Sets.svg


https://upload.wikimedia.org/wikipedia/commons/a/a2/Directed_set%252C_but_no_join_semi-lattice.png

https://upload.wikimedia.org/wikipedia/commons/a/a2/Directed_set%252C_but_no_join_semi-lattice.png

//commons.wikimedia.org/wiki/User:Jochen_Burghardt

https://upload.wikimedia.org/wikipedia/commons/0/03/Dynkin_Diagram_Isomorphisms.svg

https://upload.wikimedia.org/wikipedia/commons/0/03/Dynkin_Diagram_Isomorphisms.svg

//commons.wikimedia.org/wiki/File:Connected_Dynkin_Diagrams.svg

//commons.wikimedia.org/wiki/File:Connected_Dynkin_Diagrams.svg

//commons.wikimedia.org/wiki/User:Nonenmac

https://upload.wikimedia.org/wikipedia/commons/3/35/E-to-the-i-pi.svg

https://upload.wikimedia.org/wikipedia/commons/f/f1/EANear_and_eAMer_Supercategories.png

https://upload.wikimedia.org/wikipedia/commons/f/f1/EANear_and_eAMer_Supercategories.png


https://upload.wikimedia.org/wikipedia/en/f/f2/Edit-clear.svg

http://tango.freedesktop.org/Tango_Desktop_Project

http://tango.freedesktop.org/The_People

https://upload.wikimedia.org/wikipedia/commons/c/cf/Even_and_odd_antisymmetric_relation.png

https://upload.wikimedia.org/wikipedia/commons/c/cf/Even_and_odd_antisymmetric_relation.png

https://upload.wikimedia.org/wikipedia/en/4/48/Folder_Hexagonal_Icon.svg

https://upload.wikimedia.org/wikipedia/commons/c/cb/Frigyes_Riesz.jpeg

https://upload.wikimedia.org/wikipedia/commons/b/b0/Graph_of_non-injective%252C_non-surjective_function_%2528red%2529_and_of_bijective_function_%2528green%2529.gif




https://upload.wikimedia.org/wikipedia/en/5/59/GreaterThan.png

https://upload.wikimedia.org/wikipedia/en/0/0b/GreaterThanOrEqualTo.png

https://upload.wikimedia.org/wikipedia/commons/e/ea/Hasse_diagram_of_powerset_of_3.svg

https://upload.wikimedia.org/wikipedia/commons/e/ea/Hasse_diagram_of_powerset_of_3.svg

//commons.wikimedia.org/w/index.php?title=User:KSmrq&action=edit&redlink=1

https://upload.wikimedia.org/wikipedia/commons/9/9e/Hasse_diagram_of_powerset_of_3_no_greatest_or_least.svg

https://upload.wikimedia.org/wikipedia/commons/9/9e/Hasse_diagram_of_powerset_of_3_no_greatest_or_least.svg

//commons.wikimedia.org/wiki/File:Hasse_diagram_of_powerset_of_3.svg

//commons.wikimedia.org/wiki/File:Hasse_diagram_of_powerset_of_3.svg

//commons.wikimedia.org/wiki/User:Fibonacci

//commons.wikimedia.org/w/index.php?title=User:KSmrq&action=edit&redlink=1

https://upload.wikimedia.org/wikipedia/commons/a/a2/Hypostasis-diagram.png

//en.wikipedia.org/wiki/User:Drini

https://upload.wikimedia.org/wikipedia/commons/e/e6/Infinite_lattice_of_divisors.svg

//en.wikipedia.org/wiki/de:Datei:Verband_TeilerN.png

//commons.wikimedia.org/wiki/User:Watchduck

https://upload.wikimedia.org/wikipedia/commons/0/02/Injection.svg

https://upload.wikimedia.org/wikipedia/commons/d/df/Injective_composition.svg


• File:LampFlowchart.svg Source: https://upload.wikimedia.org/wikipedia/commons/9/91/LampFlowchart.svg License: CC-BY-SA-3.0 Contributors: vector version of Image:LampFlowchart.png Original artist: svg by Booyabazooka

• File:Lexicographic_order_on_pairs_of_natural_numbers.svg Source: https://upload.wikimedia.org/wikipedia/commons/8/8a/Lexicographic_order_on_pairs_of_natural_numbers.svg License: CC BY-SA 3.0 Contributors: Own work Original artist: Jochen Burghardt

• File:Logic_portal.svg Source: https://upload.wikimedia.org/wikipedia/commons/7/7c/Logic_portal.svg License: CC BY-SA 3.0 Con-tributors: Own work Original artist: Watchduck (a.k.a. Tilman Piesk)

• File:Mergefrom.svg Source: https://upload.wikimedia.org/wikipedia/commons/0/0f/Mergefrom.svg License: Public domain Contribu-tors: ? Original artist: ?

• File:Minimally_Descriptive_Near_Sets.svg Source: https://upload.wikimedia.org/wikipedia/commons/3/32/Minimally_Descriptive_Near_Sets.svg License: CC BY-SA 3.0 Contributors: Own work Original artist: ChristopherJamesHenry

• File:Monotonic_but_nonhomomorphic_map_between_lattices.gif Source: https://upload.wikimedia.org/wikipedia/commons/8/8c/Monotonic_but_nonhomomorphic_map_between_lattices.gif License: CC BY-SA 3.0 Contributors: Own work Original artist: JochenBurghardt

• File:N-Quadrat,_gedreht.svg Source: https://upload.wikimedia.org/wikipedia/commons/1/13/N-Quadrat%2C_gedreht.svgLicense: CC0Contributors: Own work Original artist: Mini-floh

• File:NearImages.svg Source: https://upload.wikimedia.org/wikipedia/commons/0/0d/NearImages.svg License: Public domain Contrib-utors: Own work Original artist: NearSetAccount

• File:Near_gui.jpg Source: https://upload.wikimedia.org/wikipedia/commons/f/f3/Near_gui.jpg License: CC BY-SA 3.0 Contributors:Own work Original artist: NearSetAccount

• File:Non-surjective_function2.svg Source: https://upload.wikimedia.org/wikipedia/commons/f/f0/Non-surjective_function2.svg Li-cense: CC BY-SA 3.0Contributors: http://en.wikipedia.org/wiki/File:Non-surjective_function.svgOriginal artist: original version: Maschen,the correction: raffamaiden

• File:Normal_distribution_pdf.svg Source: https://upload.wikimedia.org/wikipedia/commons/f/fb/Normal_distribution_pdf.svgLicense:CC-BY-SA-3.0 Contributors: ? Original artist: ?

• File:Nuvola_apps_edu_mathematics_blue-p.svg Source: https://upload.wikimedia.org/wikipedia/commons/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg License: GPL Contributors: Derivative work from Image:Nuvola apps edu mathematics.png and Image:Nuvolaapps edu mathematics-p.svg Original artist: David Vignoni (original icon); Flamurai (SVG convertion); bayo (color)

• File:POV-Ray-Dodecahedron.svg Source: https://upload.wikimedia.org/wikipedia/commons/a/a4/Dodecahedron.svg License: CC-BY-SA-3.0 Contributors: Vectorisation of Image:Dodecahedron.jpg Original artist: User:DTR

• File:Partial_function.svg Source: https://upload.wikimedia.org/wikipedia/commons/c/cd/Partial_function.svg License: Public domainContributors: ? Original artist: ?

• File:People_icon.svg Source: https://upload.wikimedia.org/wikipedia/commons/3/37/People_icon.svgLicense: CC0Contributors: Open-Clipart Original artist: OpenClipart

• File:Permutohedron.svg Source: https://upload.wikimedia.org/wikipedia/commons/3/3e/Permutohedron.svg License: Public domainContributors: Own work Original artist: David Eppstein

• File:Portal-puzzle.svg Source: https://upload.wikimedia.org/wikipedia/en/f/fd/Portal-puzzle.svg License: Public domain Contributors:? Original artist: ?

• File:Poset6.jpg Source: https://upload.wikimedia.org/wikipedia/en/1/12/Poset6.jpg License: PD Contributors: ? Original artist: ?• File:Proximity_System.png Source: https://upload.wikimedia.org/wikipedia/commons/c/cd/Proximity_System.png License: CC BY-

SA 3.0 Contributors: Own work Original artist: ChristopherJamesHenry• File:Question_book-new.svg Source: https://upload.wikimedia.org/wikipedia/en/9/99/Question_book-new.svg License: Cc-by-sa-3.0

Contributors:Created from scratch in Adobe Illustrator. Based on Image:Question book.png created by User:Equazcion Original artist:Tkgd2007

• File:Relación_de_dependencia.svg Source: https://upload.wikimedia.org/wikipedia/commons/9/99/Relaci%C3%B3n_de_dependencia.svg License: GFDL Contributors: Own work Original artist: Dnu72

• File:Rubik’{}s_cube_v3.svg Source: https://upload.wikimedia.org/wikipedia/commons/b/b6/Rubik%27s_cube_v3.svg License: CC-BY-SA-3.0 Contributors: Image:Rubik’{}s cube v2.svg Original artist: User:Booyabazooka, User:Meph666 modified by User:Niabot

• File:Sample_EF-space.svg Source: https://upload.wikimedia.org/wikipedia/commons/b/bf/Sample_EF-space.svg License: CC BY-SA3.0 Contributors: Own work Original artist: ChristopherJamesHenry

• File:Semiorder.svg Source: https://upload.wikimedia.org/wikipedia/commons/3/33/Semiorder.svg License: Public domain Contribu-tors: Own work Original artist: David Eppstein

• File:Series-parallel_partial_order.svg Source: https://upload.wikimedia.org/wikipedia/commons/c/c4/Series-parallel_partial_order.svg License: Public domain Contributors: Own work Original artist: David Eppstein

• File:Set_partitions_5;_matrices.svg Source: https://upload.wikimedia.org/wikipedia/commons/b/b4/Set_partitions_5%3B_matrices.svg License: CC BY 3.0 Contributors: Own work Original artist: Watchduck (a.k.a. Tilman Piesk)

• File:Software_spanner.png Source: https://upload.wikimedia.org/wikipedia/commons/8/82/Software_spanner.png License: CC-BY-SA-3.0 Contributors: Transferred from en.wikipedia; transfer was stated to be made by User:Rockfang. Original artist: Original uploaderwas CharlesC at en.wikipedia

• File:Strict_product_order_on_pairs_of_natural_numbers.svg Source: https://upload.wikimedia.org/wikipedia/commons/8/88/Strict_product_order_on_pairs_of_natural_numbers.svg License: CC BY-SA 3.0 Contributors: Own work Original artist: Jochen Burghardt

https://upload.wikimedia.org/wikipedia/commons/9/91/LampFlowchart.svg

//commons.wikimedia.org/wiki/File:LampFlowchart.png

//commons.wikimedia.org/wiki/User:Booyabazooka

https://upload.wikimedia.org/wikipedia/commons/8/8a/Lexicographic_order_on_pairs_of_natural_numbers.svg

https://upload.wikimedia.org/wikipedia/commons/8/8a/Lexicographic_order_on_pairs_of_natural_numbers.svg


https://upload.wikimedia.org/wikipedia/commons/7/7c/Logic_portal.svg


https://upload.wikimedia.org/wikipedia/commons/0/0f/Mergefrom.svg

https://upload.wikimedia.org/wikipedia/commons/3/32/Minimally_Descriptive_Near_Sets.svg

https://upload.wikimedia.org/wikipedia/commons/3/32/Minimally_Descriptive_Near_Sets.svg


https://upload.wikimedia.org/wikipedia/commons/8/8c/Monotonic_but_nonhomomorphic_map_between_lattices.gif

https://upload.wikimedia.org/wikipedia/commons/8/8c/Monotonic_but_nonhomomorphic_map_between_lattices.gif



https://upload.wikimedia.org/wikipedia/commons/1/13/N-Quadrat%252C_gedreht.svg

//commons.wikimedia.org/wiki/User:Mini-floh

https://upload.wikimedia.org/wikipedia/commons/0/0d/NearImages.svg

//commons.wikimedia.org/w/index.php?title=User:NearSetAccount&action=edit&redlink=1

https://upload.wikimedia.org/wikipedia/commons/f/f3/Near_gui.jpg


https://upload.wikimedia.org/wikipedia/commons/f/f0/Non-surjective_function2.svg

http://en.wikipedia.org/wiki/File:Non-surjective_function.svg

https://upload.wikimedia.org/wikipedia/commons/f/fb/Normal_distribution_pdf.svg

https://upload.wikimedia.org/wikipedia/commons/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg

https://upload.wikimedia.org/wikipedia/commons/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg

//commons.wikimedia.org/wiki/File:Nuvola_apps_edu_mathematics.png

//commons.wikimedia.org/wiki/File:Nuvola_apps_edu_mathematics-p.svg

//commons.wikimedia.org/wiki/File:Nuvola_apps_edu_mathematics-p.svg

//commons.wikimedia.org/wiki/User:Flamurai

//commons.wikimedia.org/wiki/User:Bayo

https://upload.wikimedia.org/wikipedia/commons/a/a4/Dodecahedron.svg

//commons.wikimedia.org/wiki/File:Dodecahedron.jpg

//commons.wikimedia.org/wiki/User:DTR

https://upload.wikimedia.org/wikipedia/commons/c/cd/Partial_function.svg

https://upload.wikimedia.org/wikipedia/commons/3/37/People_icon.svg

https://upload.wikimedia.org/wikipedia/commons/3/3e/Permutohedron.svg


https://upload.wikimedia.org/wikipedia/en/f/fd/Portal-puzzle.svg

https://upload.wikimedia.org/wikipedia/en/1/12/Poset6.jpg

https://upload.wikimedia.org/wikipedia/commons/c/cd/Proximity_System.png


https://upload.wikimedia.org/wikipedia/en/9/99/Question_book-new.svg

//en.wikipedia.org/wiki/File:Question_book.png

//en.wikipedia.org/wiki/User:Equazcion

//en.wikipedia.org/wiki/User:Tkgd2007

https://upload.wikimedia.org/wikipedia/commons/9/99/Relaci%25C3%25B3n_de_dependencia.svg

https://upload.wikimedia.org/wikipedia/commons/9/99/Relaci%25C3%25B3n_de_dependencia.svg

//commons.wikimedia.org/wiki/User:Dnu72

https://upload.wikimedia.org/wikipedia/commons/b/b6/Rubik%2527s_cube_v3.svg

//commons.wikimedia.org/wiki/File:Rubik%2527s_cube_v2.svg

//commons.wikimedia.org/wiki/User:Booyabazooka

//commons.wikimedia.org/w/index.php?title=User:Meph666&action=edit&redlink=1

//commons.wikimedia.org/wiki/User:Niabot

https://upload.wikimedia.org/wikipedia/commons/b/bf/Sample_EF-space.svg


https://upload.wikimedia.org/wikipedia/commons/3/33/Semiorder.svg


https://upload.wikimedia.org/wikipedia/commons/c/c4/Series-parallel_partial_order.svg

https://upload.wikimedia.org/wikipedia/commons/c/c4/Series-parallel_partial_order.svg


https://upload.wikimedia.org/wikipedia/commons/b/b4/Set_partitions_5%253B_matrices.svg

https://upload.wikimedia.org/wikipedia/commons/b/b4/Set_partitions_5%253B_matrices.svg


https://upload.wikimedia.org/wikipedia/commons/8/82/Software_spanner.png


//commons.wikimedia.org/wiki/User:Rockfang

//en.wikipedia.org/wiki/User:CharlesC


https://upload.wikimedia.org/wikipedia/commons/8/88/Strict_product_order_on_pairs_of_natural_numbers.svg

https://upload.wikimedia.org/wikipedia/commons/8/88/Strict_product_order_on_pairs_of_natural_numbers.svg



• File:Surjection.svg Source: https://upload.wikimedia.org/wikipedia/commons/6/6c/Surjection.svg License: Public domain Contribu-tors: ? Original artist: ?

• File:Surjective_composition.svg Source: https://upload.wikimedia.org/wikipedia/commons/a/a2/Surjective_composition.svg License:Public domain Contributors: ? Original artist: ?

• File:Surjective_function.svg Source: https://upload.wikimedia.org/wikipedia/commons/4/4d/Surjective_function.svg License: Publicdomain Contributors: Own work Original artist: Maschen

• File:Symmetry_Of_Addition.svg Source: https://upload.wikimedia.org/wikipedia/commons/4/4a/Symmetry_Of_Addition.svgLicense:Public domain Contributors: Own work Original artist: Weston.pace

• File:Text_document_with_red_question_mark.svg Source: https://upload.wikimedia.org/wikipedia/commons/a/a4/Text_document_with_red_question_mark.svg License: Public domain Contributors: Created by bdesham with Inkscape; based upon Text-x-generic.svgfrom the Tango project. Original artist: Benjamin D. Esham (bdesham)

• File:Total_function.svg Source: https://upload.wikimedia.org/wikipedia/commons/0/02/Total_function.svg License: Public domainContributors: ? Original artist: ?

• File:Transitive-closure.svg Source: https://upload.wikimedia.org/wikipedia/commons/6/60/Transitive-closure.svgLicense: GFDLCon-tributors: Own work Original artist: Anish Bramhandkar

• File:Trigonometric_functions_and_inverse.svg Source: https://upload.wikimedia.org/wikipedia/commons/0/0b/Trigonometric_functions_and_inverse.svg License: CC0 Contributors: Own work Original artist: Maschen

• File:Trigonometric_functions_and_inverse2.svg Source: https://upload.wikimedia.org/wikipedia/commons/d/d8/Trigonometric_functions_and_inverse2.svg License: CC0 Contributors: Own work Original artist: Maschen

• File:Trigonometric_functions_and_inverse3.svg Source: https://upload.wikimedia.org/wikipedia/commons/f/fa/Trigonometric_functions_and_inverse3.svg License: CC0 Contributors: Own work Original artist: Maschen

• File:Trigonometric_functions_and_inverse4.svg Source: https://upload.wikimedia.org/wikipedia/commons/2/24/Trigonometric_functions_and_inverse4.svg License: CC0 Contributors: Own work Original artist: Maschen



• File:Trigonometry_triangle.svg Source: https://upload.wikimedia.org/wikipedia/commons/7/7e/Trigonometry_triangle.svgLicense: CC-BY-SA-3.0 Contributors: Transferred from en.wikipedia to Commons. Original artist: The original uploader was Tarquin at EnglishWikipedia Later versions were uploaded by Limaner at en.wikipedia.

• File:Vector_Addition.svg Source: https://upload.wikimedia.org/wikipedia/commons/3/3d/Vector_Addition.svg License: Public do-main Contributors: ? Original artist: ?

• File:Venn_A_intersect_B.svg Source: https://upload.wikimedia.org/wikipedia/commons/6/6d/Venn_A_intersect_B.svg License: Pub-lic domain Contributors: Own work Original artist: Cepheus

• File:Visulization_of_nearness_measure.jpg Source: https://upload.wikimedia.org/wikipedia/commons/c/c9/Visulization_of_nearness_measure.jpg License: CC BY-SA 3.0 Contributors: Own work Original artist: NearSetAccount

• File:Wiki_letter_w.svg Source: https://upload.wikimedia.org/wikipedia/en/6/6c/Wiki_letter_w.svg License: Cc-by-sa-3.0 Contributors:? Original artist: ?

• File:Wiki_letter_w_cropped.svg Source: https://upload.wikimedia.org/wikipedia/commons/1/1c/Wiki_letter_w_cropped.svg License:CC-BY-SA-3.0 Contributors:

• Wiki_letter_w.svg Original artist: Wiki_letter_w.svg: Jarkko Piiroinen• File:Wikibooks-logo-en-noslogan.svg Source: https://upload.wikimedia.org/wikipedia/commons/d/df/Wikibooks-logo-en-noslogan.

svg License: CC BY-SA 3.0 Contributors: Own work Original artist: User:Bastique, User:Ramac et al.• File:Wikiversity-logo.svg Source: https://upload.wikimedia.org/wikipedia/commons/9/91/Wikiversity-logo.svg License: CC BY-SA

3.0 Contributors: Snorky (optimized and cleaned up by verdy_p) Original artist: Snorky (optimized and cleaned up by verdy_p)• File:Wiktionary-logo-en.svg Source: https://upload.wikimedia.org/wikipedia/commons/f/f8/Wiktionary-logo-en.svg License: Public

domain Contributors: Vector version of Image:Wiktionary-logo-en.png. Original artist: Vectorized by Fvasconcellos (talk · contribs),based on original logo tossed together by Brion Vibber

78.10.3 Content license• Creative Commons Attribution-Share Alike 3.0

https://upload.wikimedia.org/wikipedia/commons/6/6c/Surjection.svg

https://upload.wikimedia.org/wikipedia/commons/a/a2/Surjective_composition.svg

https://upload.wikimedia.org/wikipedia/commons/4/4d/Surjective_function.svg

//commons.wikimedia.org/wiki/User:Maschen

https://upload.wikimedia.org/wikipedia/commons/4/4a/Symmetry_Of_Addition.svg

//commons.wikimedia.org/w/index.php?title=User:Weston.pace&action=edit&redlink=1

https://upload.wikimedia.org/wikipedia/commons/a/a4/Text_document_with_red_question_mark.svg

https://upload.wikimedia.org/wikipedia/commons/a/a4/Text_document_with_red_question_mark.svg

//commons.wikimedia.org/wiki/User:Bdesham

//commons.wikimedia.org/wiki/File:Text-x-generic.svg

//commons.wikimedia.org/wiki/User:Bdesham

https://upload.wikimedia.org/wikipedia/commons/0/02/Total_function.svg

https://upload.wikimedia.org/wikipedia/commons/6/60/Transitive-closure.svg

https://upload.wikimedia.org/wikipedia/commons/0/0b/Trigonometric_functions_and_inverse.svg

https://upload.wikimedia.org/wikipedia/commons/0/0b/Trigonometric_functions_and_inverse.svg


https://upload.wikimedia.org/wikipedia/commons/d/d8/Trigonometric_functions_and_inverse2.svg

https://upload.wikimedia.org/wikipedia/commons/d/d8/Trigonometric_functions_and_inverse2.svg


https://upload.wikimedia.org/wikipedia/commons/f/fa/Trigonometric_functions_and_inverse3.svg

https://upload.wikimedia.org/wikipedia/commons/f/fa/Trigonometric_functions_and_inverse3.svg


https://upload.wikimedia.org/wikipedia/commons/2/24/Trigonometric_functions_and_inverse4.svg









https://upload.wikimedia.org/wikipedia/commons/7/7e/Trigonometry_triangle.svg


//en.wikipedia.org/wiki/User:Tarquin

//en.wikipedia.org/wiki/User:Limaner


https://upload.wikimedia.org/wikipedia/commons/3/3d/Vector_Addition.svg

https://upload.wikimedia.org/wikipedia/commons/6/6d/Venn_A_intersect_B.svg

//commons.wikimedia.org/w/index.php?title=User:Cepheus&action=edit&redlink=1

https://upload.wikimedia.org/wikipedia/commons/c/c9/Visulization_of_nearness_measure.jpg

https://upload.wikimedia.org/wikipedia/commons/c/c9/Visulization_of_nearness_measure.jpg


https://upload.wikimedia.org/wikipedia/en/6/6c/Wiki_letter_w.svg

https://upload.wikimedia.org/wikipedia/commons/1/1c/Wiki_letter_w_cropped.svg

//commons.wikimedia.org/wiki/File:Wiki_letter_w.svg

//commons.wikimedia.org/wiki/File:Wiki_letter_w.svg

//commons.wikimedia.org/wiki/User:Jarkko_Piiroinen

https://upload.wikimedia.org/wikipedia/commons/d/df/Wikibooks-logo-en-noslogan.svg

https://upload.wikimedia.org/wikipedia/commons/d/df/Wikibooks-logo-en-noslogan.svg

//commons.wikimedia.org/wiki/User:Bastique

//commons.wikimedia.org/wiki/User:Ramac

https://upload.wikimedia.org/wikipedia/commons/9/91/Wikiversity-logo.svg

//commons.wikimedia.org/w/index.php?title=User:Snorky&action=edit&redlink=1

//commons.wikimedia.org/wiki/User:Verdy_p

//commons.wikimedia.org/w/index.php?title=User:Snorky&action=edit&redlink=1

//commons.wikimedia.org/wiki/User:Verdy_p

https://upload.wikimedia.org/wikipedia/commons/f/f8/Wiktionary-logo-en.svg

//commons.wikimedia.org/wiki/File:Wiktionary-logo-en.png

//commons.wikimedia.org/wiki/User:Fvasconcellos

//commons.wikimedia.org/wiki/User_talk:Fvasconcellos

//commons.wikimedia.org/wiki/Special:Contributions/Fvasconcellos

//commons.wikimedia.org/wiki/User:Brion_VIBBER

https://creativecommons.org/licenses/by-sa/3.0/

Documents

Mathematical Relations Az