Mathematical Relations 1

Mathematical relations 1From 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 . . . . . . . . . . . . . . . . . 31.4 Philosophical Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.5 Computer Science Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Allegory (category theory) 62.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Regular categories and allegories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 Allegories of relations in regular categories . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2.2 Maps in allegories, and tabulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2.3 Unital allegories and regular categories of maps . . . . . . . . . . . . . . . . . . . . . . . 72.2.4 More sophisticated kinds of allegory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Alternatization 83.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.1.1 Alternating bilinear map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.1.2 Alternating bilinear form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.1.3 Alternating multilinear form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.1.4 Alternatization of a bilinear map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4 Ancestral relation 104.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.2 Relationship to transitive closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

i

ii CONTENTS

4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

5 Antisymmetric relation 125.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

6 Asymmetric relation 146.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

7 Better-quasi-ordering 157.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157.2 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157.3 Simpsons alternative denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157.4 Major theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

8 Bidirectional transformation 178.1 Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178.2 Vocabulary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178.3 Examples of implementations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

9 Bijection 189.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

9.2.1 Batting line-up of a baseball team . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189.2.2 Seats and students of a classroom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

9.3 More mathematical examples and some non-examples . . . . . . . . . . . . . . . . . . . . . . . . 199.4 Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199.5 Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199.6 Bijections and cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199.7 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209.8 Bijections and category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

CONTENTS iii

9.9 Generalization to partial functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209.10 Contrast with . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209.11 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209.12 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219.13 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219.14 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

10 Bijection, injection and surjection 2210.1 Injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2210.2 Surjection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2310.3 Bijection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2310.4 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2310.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

10.5.1 Injective and surjective (bijective) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2410.5.2 Injective and non-surjective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2410.5.3 Non-injective and surjective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2410.5.4 Non-injective and non-surjective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

10.6 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2410.7 Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2410.8 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2410.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2410.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

11 Homogeneous relation 2511.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

11.1.1 Is a relation more than its graph? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2511.1.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

11.2 Special types of binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2611.2.1 Difunctional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

11.3 Relations over a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2711.4 Operations on binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

11.4.1 Complement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2911.4.2 Restriction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2911.4.3 Algebras, categories, and rewriting systems . . . . . . . . . . . . . . . . . . . . . . . . . 29

11.5 Sets versus classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2911.6 The number of binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3011.7 Examples of common binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3011.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3011.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3011.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3111.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

iv CONTENTS

12 Cointerpretability 3212.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3212.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

13 Commutative property 3313.1 Common uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3313.2 Mathematical denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3313.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

13.3.1 Commutative operations in everyday life . . . . . . . . . . . . . . . . . . . . . . . . . . . 3313.3.2 Commutative operations in mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . 3313.3.3 Noncommutative operations in everyday life . . . . . . . . . . . . . . . . . . . . . . . . . 3413.3.4 Noncommutative operations in mathematics . . . . . . . . . . . . . . . . . . . . . . . . . 34

13.4 History and etymology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3513.5 Propositional logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

13.5.1 Rule of replacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3513.5.2 Truth functional connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

13.6 Set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3513.7 Mathematical structures and commutativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3513.8 Related properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

13.8.1 Associativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3513.8.2 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

13.9 Non-commuting operators in quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 3613.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3613.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3713.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

13.12.1 Books . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3713.12.2 Articles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3713.12.3 Online resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

14 Comparability 3814.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3814.2 Comparability graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3814.3 Classication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3814.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3814.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

15 Composition of relations 3915.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3915.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3915.3 Join: another form of composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3915.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3915.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

CONTENTS v

15.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

16 Congruence relation 4116.1 Basic example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4116.2 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4116.3 Relation with homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4216.4 Congruences of groups, and normal subgroups and ideals . . . . . . . . . . . . . . . . . . . . . . . 42

16.4.1 Ideals of rings and the general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4216.5 Universal algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4216.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4216.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

17 Contour set 4317.1 Formal denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

17.1.1 Contour sets of a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4317.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

17.2.1 Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4317.2.2 Economic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

17.3 Complementarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4417.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4417.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4417.6 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

18 Coreexive relation 4618.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

19 Demonic composition 4719.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4719.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

20 Dense order 4820.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4820.2 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4820.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4820.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

21 Dependence relation 4921.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4921.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

22 Dependency relation 5022.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

23 Homogeneous relation 51

vi CONTENTS

23.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5123.1.1 Is a relation more than its graph? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5123.1.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52





24 Directed set 5824.1 Equivalent denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5824.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5824.3 Contrast with semilattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5824.4 Directed subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5924.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5924.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5924.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

25 Equality (mathematics) 6025.1 Etymology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6025.2 Types of equalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

25.2.1 Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6025.2.2 Equalities as predicates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6025.2.3 Congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6025.2.4 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6025.2.5 Equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

25.3 Logical formalizations of equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6025.4 Logical formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6125.5 Some basic logical properties of equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6125.6 Relation with equivalence and isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6125.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6225.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

CONTENTS vii

26 Equipollence (geometry) 6326.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6326.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

27 Equivalence class 6427.1 Notation and formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6427.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6527.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6527.4 Graphical representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6527.5 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6527.6 Quotient space in topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6527.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6627.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6627.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6627.10Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

28 Equivalence relation 6828.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6828.2 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6828.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

28.3.1 Simple example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6828.3.2 Equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6828.3.3 Relations that are not equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

28.4 Connections to other relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6928.5 Well-denedness under an equivalence relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6928.6 Equivalence class, quotient set, partition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

28.6.1 Equivalence class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7028.6.2 Quotient set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7028.6.3 Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7028.6.4 Equivalence kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7028.6.5 Partition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

28.7 Fundamental theorem of equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7028.8 Comparing equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7128.9 Generating equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7128.10Algebraic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

28.10.1 Group theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7128.10.2 Categories and groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7228.10.3 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

28.11Equivalence relations and mathematical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7228.12Euclidean relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7328.13See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7328.14Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

viii CONTENTS

28.15References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7328.16External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

29 Euclidean relation 7529.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7529.2 Relation to transitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7529.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

30 Exceptional isomorphism 7630.1 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

30.1.1 Finite simple groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7630.1.2 Groups of Lie type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7630.1.3 Alternating groups and symmetric groups . . . . . . . . . . . . . . . . . . . . . . . . . . 7630.1.4 Cyclic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7730.1.5 Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7730.1.6 Coxeter groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

30.2 Lie theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7730.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7830.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7830.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

31 Fiber (mathematics) 7931.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

31.1.1 Fiber in naive set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7931.1.2 Fiber in algebraic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

31.2 Terminological variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7931.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

32 Finitary relation 8132.1 Informal introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8132.2 Relations with a small number of places . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8132.3 Formal denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8232.4 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8332.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8332.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8332.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8332.8 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

33 Foundational relation 8533.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8533.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

34 Homogeneous relation 8634.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

CONTENTS ix

34.1.1 Is a relation more than its graph? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8634.1.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87





35 Hypostatic abstraction 9335.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9335.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9335.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

36 Idempotence 9536.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

36.1.1 Unary operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9536.1.2 Idempotent elements and binary operations . . . . . . . . . . . . . . . . . . . . . . . . . . 9536.1.3 Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

36.2 Common examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9636.2.1 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9636.2.2 Formal languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9636.2.3 Idempotent ring elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9636.2.4 Other examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

36.3 Computer science meaning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9636.3.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

36.4 Applied examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9736.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9736.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9736.7 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9836.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

37 Idempotent relation 9937.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

x CONTENTS

38 Intransitivity 10038.1 Intransitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10038.2 Antitransitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10038.3 Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10038.4 Occurrences in preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10138.5 Likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10138.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10138.7 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

39 Inverse relation 10239.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

39.1.1 Inverse relation of a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10239.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10239.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10339.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

40 Inverse trigonometric functions 10440.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

40.1.1 Etymology of the arc- prex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10440.2 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

40.2.1 Principal values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10440.2.2 Relationships between trigonometric functions and inverse trigonometric functions . . . . . 10540.2.3 Relationships among the inverse trigonometric functions . . . . . . . . . . . . . . . . . . . 10540.2.4 Arctangent addition formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

40.3 In calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10640.3.1 Derivatives of inverse trigonometric functions . . . . . . . . . . . . . . . . . . . . . . . . 10640.3.2 Expression as denite integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10640.3.3 Innite series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10740.3.4 Indenite integrals of inverse trigonometric functions . . . . . . . . . . . . . . . . . . . . 107

40.4 Extension to complex plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10840.4.1 Logarithmic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

40.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10940.5.1 General solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10940.5.2 In computer science and engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

40.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11040.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11040.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

41 Near sets 11241.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11341.2 Nearness of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11441.3 Generalization of set intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

CONTENTS xi

41.4 Efremovi proximity space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11441.5 Visualization of EF-axiom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11541.6 Descriptive proximity space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11541.7 Proximal relator spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11641.8 Descriptive -neighbourhoods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11641.9 Tolerance near sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11741.10Tolerance classes and preclasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

41.10.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11741.11Nearness measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11841.12Near set evaluation and recognition (NEAR) system . . . . . . . . . . . . . . . . . . . . . . . . . 11941.13Proximity System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11941.14See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11941.15Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11941.16References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12041.17Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

42 Partial equivalence relation 12442.1 Properties and applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12442.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

42.2.1 Euclidean parallelism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12442.2.2 Kernels of partial functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12442.2.3 Functions respecting equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . 124

42.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12542.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

43 Partial function 12643.1 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12643.2 Total function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12643.3 Discussion and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

43.3.1 Natural logarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12743.3.2 Subtraction of natural numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12743.3.3 Bottom element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12743.3.4 In category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12743.3.5 In abstract algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

43.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12743.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

44 Partially ordered set 12944.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12944.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12944.3 Extrema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13044.4 Orders on the Cartesian product of partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . 130

xii CONTENTS

44.5 Sums of partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13044.6 Strict and non-strict partial orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13144.7 Inverse and order dual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13144.8 Mappings between partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13144.9 Number of partial orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13244.10Linear extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13244.11In category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13244.12Partial orders in topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13244.13Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13244.14See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13244.15Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13344.16References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13344.17External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

45 Preorder 13445.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13445.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13445.3 Uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13545.4 Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13545.5 Number of preorders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13645.6 Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13645.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13645.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

46 Prewellordering 13746.1 Prewellordering property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

46.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13746.1.2 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137


47 Propositional function 13947.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

48 Quasi-commutative property 14048.1 Applied to matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14048.2 Applied to functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14048.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14048.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

49 Quasitransitive relation 14149.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14149.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

CONTENTS xiii

49.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14149.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14149.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

50 Quotient by an equivalence relation 14250.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14250.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14250.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14250.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

51 Rational consequence relation 14351.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14351.2 Uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

51.2.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14351.2.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

51.3 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14451.4 Rational consequence relations via atom preferences . . . . . . . . . . . . . . . . . . . . . . . . . 144

51.4.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14451.5 The representation theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

51.5.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14451.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

52 Reduct 14552.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14552.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14552.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

53 Reexive closure 14653.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14653.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14653.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

54 Reexive relation 14754.1 Related terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14754.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14754.3 Number of reexive relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14854.4 Philosophical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14854.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14854.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14854.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14854.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

55 Relation algebra 14955.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

xiv CONTENTS

55.1.1 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14955.2 Expressing properties of binary relations in RA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15055.3 Expressive power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

55.3.1 Q-Relation Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15055.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15055.5 Historical remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15155.6 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15155.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15155.8 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15155.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15155.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

56 Relation construction 15356.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

57 Representation (mathematics) 15457.1 Representation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15457.2 Other examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

57.2.1 Graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15457.2.2 Order theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15457.2.3 Polysemy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154


58 Semiorder 15658.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15658.2 Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15658.3 Other results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15758.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15758.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15758.6 Additional reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

59 Separoid 15959.1 The axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15959.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15959.3 The basic lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15959.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

60 Sequential composition 16160.1 Essential features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16160.2 Mathematics of processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

60.2.1 Parallel composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16160.2.2 Communication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

CONTENTS xv

60.2.3 Sequential composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16260.2.4 Reduction semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16260.2.5 Hiding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16260.2.6 Recursion and replication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16260.2.7 Null process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

60.3 Discrete and continuous process algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16260.4 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16260.5 Current research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16360.6 Software implementations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16360.7 Relationship to other models of concurrency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16360.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16460.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16460.10Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

61 Series-parallel partial order 16561.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16561.2 Forbidden suborder characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16661.3 Order dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16661.4 Connections to graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16661.5 Computational complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16661.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16761.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16761.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

62 Surjective function 16862.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16862.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16862.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

62.3.1 Surjections as right invertible functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16962.3.2 Surjections as epimorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17062.3.3 Surjections as binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17062.3.4 Cardinality of the domain of a surjection . . . . . . . . . . . . . . . . . . . . . . . . . . . 17062.3.5 Composition and decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17062.3.6 Induced surjection and induced bijection . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

62.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17062.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17062.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

63 Symmetric closure 17263.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17263.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17263.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

xvi CONTENTS

64 Symmetric relation 17364.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

64.1.1 In mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17364.1.2 Outside mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

64.2 Relationship to asymmetric and antisymmetric relations . . . . . . . . . . . . . . . . . . . . . . . 17364.3 Additional aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17464.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

65 Ternary equivalence relation 17565.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17565.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

66 Ternary relation 17666.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

66.1.1 Binary functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17666.1.2 Cyclic orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17666.1.3 Betweenness relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17666.1.4 Congruence relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17666.1.5 Typing relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

66.2 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

67 Tolerance relation 17867.1 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

68 Total order 17968.1 Strict total order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17968.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17968.3 Further concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

68.3.1 Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18068.3.2 Lattice theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18068.3.3 Finite total orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18068.3.4 Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18068.3.5 Order topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18068.3.6 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18068.3.7 Sums of orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

68.4 Orders on the Cartesian product of totally ordered sets . . . . . . . . . . . . . . . . . . . . . . . . 18168.5 Related structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18168.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18168.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18168.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

69 Total relation 18369.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

CONTENTS xvii

69.2 Properties and related notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18369.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18369.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

70 Transitive closure 18470.1 Transitive relations and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18470.2 Existence and description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18470.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18570.4 In graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18570.5 In logic and computational complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18570.6 In database query languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18570.7 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18570.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18670.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18670.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

71 Transitive reduction 18771.1 In directed acyclic graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18771.2 In graphs with cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18771.3 Computational complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18871.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18871.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18871.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

72 Transitive relation 19072.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19072.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19072.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

72.3.1 Closure properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19072.3.2 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19072.3.3 Properties that require transitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

72.4 Counting transitive relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19172.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19172.6 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

72.6.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19172.6.2 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

72.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

73 Trichotomy (mathematics) 19273.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19273.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

74 Unimodality 193

xviii CONTENTS

74.1 Unimodal probability distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19374.1.1 Other denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19474.1.2 Uses and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19474.1.3 Gauss inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19474.1.4 VysochanskiPetunin inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19474.1.5 Mode, median and mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19474.1.6 Skewness and kurtosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

74.2 Unimodal function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19474.3 Other extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19574.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19574.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

75 Weak ordering 19675.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19675.2 Axiomatizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

75.2.1 Strict weak orderings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19775.2.2 Total preorders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19775.2.3 Ordered partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19775.2.4 Representation by functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

75.3 Related types of ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19875.4 All weak orders on a nite set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

75.4.1 Combinatorial enumeration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19875.4.2 Adjacency structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

75.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19975.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

76 Well-founded relation 20176.1 Induction and recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20176.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20276.3 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20276.4 Reexivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20276.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

77 Well-order 20377.1 Ordinal numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20377.2 Examples and counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

77.2.1 Natural numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20377.2.2 Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20477.2.3 Reals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

77.3 Equivalent formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20477.4 Order topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20477.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

CONTENTS xix

77.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

78 Well-quasi-ordering 20678.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20678.2 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20678.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20678.4 Wqos versus well partial orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20778.5 Innite increasing subsequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20778.6 Properties of wqos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20778.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20778.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20878.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20878.10Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 209

78.10.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20978.10.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21678.10.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

Chapter 1

Accessibility relation

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

1.1 Description of TermsA '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 iscold' or false that 'the room is cold.'Generally, commands, beliefs and sentences about prob-abilities aren't judged as true or false. 'Inhale and exhale'is therefore not a statement in logic because it is a com-mand and cannot be true or false, although a person canobey or refuse that command. 'I believe I can y or I can'ty' isn't taken as a statement of truth or falsity, becausebeliefs don't say anything about the truth or falsity of theparts of the entire 'and' or 'or' statement and therefore theentire 'and' or 'or' statement.A 'possible world' is any possible situation. In everycase, a 'possible world' is contrasted with an actual sit-uation. Earth one minute from now is a 'possible world.'The earth as it actually is also a 'possible world.' Hencethe oddity of and controversy in contrasting a 'possible'world with an 'actual world' (earth is necessarily possi-ble). In logic, 'worlds are described as a non-empty set,where the set could consist of anything, depending onwhat the statement says.'Modal Logic' is a description of the reasoning in makingstatements about 'possibility' or 'necessity.' 'It is possiblethat it rains tomorrow' is a statement in modal logic, be-cause it is a statement about possibility. 'It is necessarythat it rains tomorrow' also counts as a statement in modallogic, because it is a statement about 'necessity.' Thereare at least six logical axioms or principles that show whatpeople mean whenever they make statements about 'ne-cessity' or 'possibility' (described below). For a detailedexplanation on modal logic, see here.As described in greater detail below:Necessarily p means that p is true at every 'possibleworld' w such that R(w; w):

Possibly p means that p is true at some possible worldw 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 onthe meanings of words, laws of logic, or experience (ob-servation, hearing, etc.).'Formal Semantics refers to the meaning of statementswritten in symbols. The sentence (p_q)! (p_ q), for example, is a statement about 'necessity' in 'formalsemantics.' It has a meaning that can be represented bythe symbol R .The 'accessibility relation' is a relationship between two'possible worlds.' More preciselyplease clarify denition, the'accessibility relation' is the idea that modal statements,like 'its possible that it rains tomorrow,' may not take thesame truth-value in all 'possible worlds.' On earth, thestatement could be true or false. By contrast, in a planetwhere water is non-existent, this statement will always befalse.Due to the diculty in judging if a modal statement istrue in every 'possible world,' logicians have derived cer-tain axioms or principles that show on what basis anystatement is true in any 'possible world.' These axiomsdescribing the relationship between 'possible worlds isthe 'accessibility relation' in detail.Put another way, these modal axioms describe in de-tail the 'accessibility relation,' R between two 'worlds.'That relation, R symbolizes that from any given 'possi-ble world' some other 'possible worlds may be accessible,and others may not be.The 'accessibility relation' has important uses in boththe formal/theoretical aspects of modal logic (theoriesabout 'modal logic'). It also has applications to a vari-ety of disciplines including epistemology (theories abouthow people know something is true or false), metaphysics(theories about reality), value theory (theories aboutmorality and ethics), and computer science (theoriesabout programmatic manipulation of data).

1

2 CHAPTER 1. ACCESSIBILITY RELATION

1.2 Basic Review of (Propositional)Modal Logic

The reasoning behind the 'accessibility relation' uses thebasics of 'propositional modal logic' (see modal logic fora detailed discussion). 'Propositional modal logic' is tra-ditional propositional logic with the addition of two keyunary operators: symbolizes the phrase 'It is necessary that...' symbolizes the phrase 'It is possible that...'These operators can be attached to a single sentence toform a new compound sentence.For example, can be attached to a sentence such as 'Iwalk outside.' The new sentence would look like: 'Iwalk outside.' The entire new sentence would thereforeread: 'It is necessary that I walk outside.'But the symbol A can be used to stand for any sentenceinstead of writing out entire sentences. So any sentencesuch as 'I walk outside' or 'I walk outside and I lookaround' are symbolized by A .Thus for any sentenceA (simple or compound), the com-pound sentences A and A can be formed. Sentencessuch as 'It is necessary that I walk outside' or 'It is possi-ble that I walk outside' would therefore look like: AA .However, the symbols p , q can also be used to stand forany 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' wouldlook like: q . The sentence 'It is possible that I walkoutside' would look like: q .Six Basic Axioms of Modal Logic:There are at least six basic axioms or principles of almostall modal logics or steps in thinking/reasoning. The rsttwo hold in all regular modal logics, and the last holds inall normal modal logics.1st Modal Axiom:

p$ ::p (Duality)

The double arrow stands symbolizes 'if and only if,' 'nec-essary and sucient' conditions. A 'necessary' conditionis something that must be the case for something else.Being literate, for instance, is a 'necessary' condition forreading about the 'accessibility relation.' A 'sucientcondition' a condition that is good enough for somethingelse. Being literate, for instance, is a 'sucient' condi-tion for learning about the accessibility relation.' In otherwords, its good enough to be literate in order to learnabout the 'accessibility relation,' however it may not be'necessary' because the relation could be learned in dif-ferent ways (like through speech). Aside from 'necessaryand sucient,' the double arrow represents equivalence

between the meaning of two statements, the statement tothe left and the statement to the right of the double arrow.The half square symbols before the diamond and p sym-bol in the sentence ' p$ ::p ' stand for 'it is not thecase, or 'not.'The p symbol stands for any statement such as 'I walkoutside.' Therefore it could also stand for 'The apple isRed.'Example 1:The rst principle says that any statement involving 'ne-cessity' on the left side of the double arrow is equivalentto the statement about the negation of 'possibility' on theright.So using the symbols and their meaning, the rst modalaxiom, p$ ::p could stand for: 'Its necessary thatI walk outside if and only if its not possible that it is notthe case that I walk outside.'And when I say that 'Its necessary that I walk outside,'this is the same as saying that 'Its not possible that it isnot the case that I walk outside.' Furthermore, when Isay that 'Its not possible that it is not the case that I walkoutside,' this is the same as saying that 'Its necessary thatI walk outside.'Example 2:p stands for 'The apple is red.'So using the symbols and their meaning described above,the rst modal axiom, p$ ::p could stand for: 'Itsnecessary that the apple is red if and only if its not pos-sible that it is not the case that the apple is red.'And when I say that 'Its necessary that the apple is red,'this is the same as saying that 'Its not possible that it isnot the case that the apple is red.' Furthermore, when Isay that 'Its not possible that it is not the case that theapple is red,' this is the same as saying that 'Its necessarythat 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 sameas the statement about the negation of 'necessity' on theright.p stands for 'Spring has not arrived.'Using the symbols and their meaning described above,the second modal axiom, p $ ::p could stand for:'Its possible that Spring has not arrived if and only if it isnot the case that it is necessary that it is not the case thatSpring has not arrived.'Essentially, the second axiom says that any statementabout possibility called 'X' is the same as a negation or

1.3. THE FOUR TYPES OF THE 'ACCESSIBILITY RELATION' IN FORMAL SEMANTICS 3

denial of a dierent statement about necessity 'Y.' Thestatement about necessity shows the denial of the sameoriginal statement 'X.'The other axioms can be read and interpreted in the sameway, by substituting letters p for any statement and fol-lowing the reasoning. Brackets in a symbolized sentencemean that anything inside the brackets counts as a wholesentence. Any symbol before the brackets therefore ap-plies to the sentence as a whole, not just the letters or anindividual sentence.An arrow stands for then where the left statement beforethe 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 operatorsare controversial and not widely agreed upon. Here arethe most 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 tradi-tional names of these axioms (or principles).According to the traditional 'possible worlds semantics ofmodal logic, the compound sentences that are formed outof the modal operators are to be interpreted in terms ofquantication 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 istrue (symmetry, transitive property, etc.) in all 'possibleworlds is the 'accessibility relation.'The relation of accessibility can now be dened as an (un-interpreted) relation R(w1; w2) that holds between 'pos-sible worlds w1 and w2 only when w2 is accessiblefrom w1 .Additionally, to make things more specic, any formula,axiom like (p _ q) ! (p _ q) can be translatedinto a formula of rst-order logic through standard trans-lation. That rst-order logic formula or sentence makesthe meaning of the boxes and diamonds in modal logicexplicit.

1.3 The Four Types of the 'Acces-sibility Relation' in Formal Se-mantics

'Formal semantics studies the meaning of statementswritten in symbols. The sentence (p_q)! (p_ q), for example, is a statement about 'necessity' in 'formalsemantics.' It has a meaning that can be represented bythe symbol R , where R takes the form of the 'necessityrelation' described below.So, the 'accessibility relation,' R can take on at least fourforms, 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 dened 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 possibleworld w such that R(w; w) .

The four types of R will be a variation of these two gen-eral types. They will specify on what conditions a state-ment is true either in every possible world, or some pos-sible. The four specic types of R are:Reexive, or *Axiom (T) above:If R is reexive, every world is accessible to itself. Re-exivity guarantees that any world at which A is true isa world from which there is an accessible world at whichA is true, and thus A is possible at worlds where its true,which isn't necessarily the case in worlds that aren't acces-sible to themselves. Without the reexivity condition, Acan be necessary at a world where its false, if that worldisn't accessible to itself; thus axiom Tthat A at aworld implies A is true at that worldfollows from re-exivity.Transitive, or *Axiom (4) above:If R is transitive, any world accessible to any world w0accessible to world w is also accessible to w . Transi-tively, A is true at a world w only when A is true atevery world w0 accessible to w , including every worldw00 accessible to any w0 , and every world accessible toany w00 , etc., so when A is true at w , its also trueat every w0 and every w00 , etc., which means A isalso true at w , which is axiom 4.Euclidean or *Axiom (5) above:If R is euclidean, any two worlds accessible to a givenworld are accessible to each other. A is true at a worldw if and only if, for every world w0 accessible to w ,there is a world w00 accessible to w0 at which A is true.If A is true at a world w0 accessible to w , then if thatworld is accessible to every other world accessible to w ,

4 CHAPTER 1. ACCESSIBILITY RELATION

it will be true that for every world accessible to w thereis an accessible world ( w0 ) at which A is true, so A istrue at all worlds accessible to w . The euclidean propertythus entails that A implies A , which is axiom 5.Symmetric or *Axiom (B) above:If R is symmetric, then if world w0 is accessible toworld w , w is accessible to w0 . If A is true at w ,then at every w0 accessible to w , there is a world ( w )accessible to w0 at which A is true, so A is possible atall w0 , and thus its necessary at w that A is possible,which is axiom B.

1.4 Philosophical ApplicationsOne of the applications of 'possible worlds semanticsand the 'accessibility relation' is to physics. Instead ofjust talking generically about 'necessity (or logical neces-sity),' the relation in physics deals with 'nomological ne-cessity.' The fundamental translational schema (TS) de-scribed earlier can be exemplied as follows for physics:

(TSN) P is nomologically necessary means that Pis true at all possible worlds that are nomologicallyaccessible from the actual world. In other words, Pis true at all possible worlds that obey the physicallaws of the actual world.

The interesting thing to observe is that instead of hav-ing to ask, now, Does nomological necessity satisfy theaxiom (5)?", that is, Is something that is nomologicallypossible nomologically necessarily possible?", we can askinstead: Is the nomological accessibility relation eu-clidean?" And dierent theories of the nature of phys-ical laws will result in dierent answers to this question.(Notice however that if the objection raised earlier is true,each dierent theory of the nature of physical laws wouldbe 'possible' and 'necessary,' since the euclidean conceptdepends on the idea about 'possibility' and 'necessity').The theory of Lewis, for example, is asymmetric. Hiscounterpart theory also requires an intransitive relation ofaccessibility because it is based on the notion of similar-ity and similarity is generally intransitive. For example, apile of straw with one less handful of straw may be similarto the whole pile but a pile with two (or more) less hand-fuls may not be. So x can be necessarily P without xbeing necessarily necessarily P . On the other hand, SaulKripke has an account of de re modality which is basedon (metaphysical) identity across worlds and is thereforetransitive.Another interpretation of the 'accessibility relation' witha physical meaning was given in Gerla 1987 where theclaim is possible P in the world w00 is interpretedas it is possible to transform w into a world in whichP is true. So, the properties of the modal operatorsdepend on the algebraic properties of the set of admissibletransformations.

There are other applications of the 'accessibility relation'in philosophy. In epistemology, one can, instead of talk-ing about nomological accessibility, talk about epistemicaccessibility. A world w0 is epistemically accessible fromw for an individual I in w if and only if I does notknow something which would rule out the hypothesis thatw0 = w . We can ask whether the relation is transitive. IfI knows nothing that rules out the possibility thatw0 = wand knows nothing that rules the possibility thatw00 = w0, it does not follow that I knows nothing which rules outthe hypothesis that w00 = w . To return to our earlier ex-ample, one may not be able to distinguish a pile of sandfrom the same pile with one less handful and one may notbe able to distinguish the pile with one less handful fromthe same pile with two less handfuls of sand, but one maystill be able to distinguish the original pile from the pilewith two less handfuls of sand.Yet another example of the use of the 'accessibility re-lation' is in deontic logic. If we think of obligatorinessas truth in all morally perfect worlds, and permissibil-ity as truth in some morally perfect world, then we willhave to restrict out universe to include only morally per-fect worlds. But, in that case, we will have left out theactual world. A better alternative would be to include allthe metaphysically possible worlds but restrict the 'acces-sibility relation' to morally perfect worlds. Transitivityand the euclidean property will hold, but reexivity andsymmetry will not.

1.5 Computer Science ApplicationsIn modeling a computation, a 'possible world' can bea possible computer state. Given the current computerstate, you might dene the accessible possible worlds tobe all future possible computer states, or to be all possibleimmediate next computer states (assuming a discretecomputer). Either choice denes a particular 'accessibil-ity relation' giving rise to a particular modal logic suitedspecically for theorems about the computation.

1.6 See also Modal logic Possible worlds Propositional attitude Modal depth

1.7 References Gerla, G.; Transformational sem

Documents

Mathematical Relations 1