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CAS 738: AlgebraicMethods
in SoftwareEngineering
and ComputerScience
Dr. R. Khedri
Outline
Introduction
Binary relations
Relation algebras
Atom structures ofRAs
Examples of relationalgebra
(Slide 1 of 38)
CAS 738: Algebraic Methodsin Software Engineeringand Computer Science
Dr. Ridha Khedri
Department of Computing and Software, McMaster UniversityCanada L8S 4L7, Hamilton, Ontario
CAS 738: AlgebraicMethods
in SoftwareEngineering
and ComputerScience
Dr. R. Khedri
Outline
Introduction
Binary relations
Relation algebras
Atom structures ofRAs
Examples of relationalgebra
(Slide 2 of 38)
1 Introduction2 Binary relations
Proper relation algebrasSquare proper relation algebras
3 Relation algebras
Definition of relation algebras
Peircean law4
Atom structures of relation algebras
Boolean algebras with operator
Atom structures of RAs
Consistent and forbidden triples of atoms
Representations of relation algebras5 Examples of relation algebra
CAS 738: AlgebraicMethods
in SoftwareEngineering
and ComputerScience
Dr. R. Khedri
Outline
Introduction
Binary relations
Relation algebras
Atom structures ofRAs
Examples of relationalgebra
(Slide 3 of 38)Relation Algebras Introduction
We start with a very basic mathematical object, arelation of a certain rank (arity)
We actually look at a field of relations:
a set of relations, over some fixed domainclosed under certain concrete operations: union,intersection, complement of a relation (relative to somebiggest relation over the domain)
So we have a field of sets
We need other operations pertaining to the relationalproperties
identity relation, the converse of a relation, and thecomposition of two binary relationsa field of binary relations should then include theidentity and be closed under conversion andcomposition
CAS 738: AlgebraicMethods
in SoftwareEngineering
and ComputerScience
Dr. R. Khedri
Outline
Introduction
Binary relations
Relation algebras
Atom structures ofRAs
Examples of relationalgebra
(Slide 4 of 38)Relation Algebras Introduction
Then, we write out some axioms for these operations
The set of axioms ⊇ the set of axioms for booleanalgebra ∪ a set of other axioms involving therelational operators
The axioms will almost invariably be valid over fields ofrelations of the appropriate type
However, the question of completeness is more tricky
Let us call an abstract structure that obeys theseaxioms an algebraThe question of completeness can now be thought of inanother way: is any arbitrary algebra (isomorphic toconcrete structures discussed) isomorphic to a field ofrelations of the appropriate type?An isomorphism from an algebra to a field of relationsis called a representation
CAS 738: AlgebraicMethods
in SoftwareEngineering
and ComputerScience
Dr. R. Khedri
Outline
Introduction
Binary relations
Relation algebras
Atom structures ofRAs
Examples of relationalgebra
(Slide 5 of 38)Relation Algebras Introduction
What are the advantages of working with the algebrasrather than the concrete structures?
The technique of focusing our whole attention on asmall number of properties to the exclusion of allothers is quite general in scientific analysis
Because we are ignoring many of the concrete featuresof our structures, we may find that our approach ismore general
Algebra is a very thoroughly studied discipline andthere is a range of methods and results that we canapply to the study of relations
Universal algebra and model theory have techniquesand theorems that can be exploited fruitfully here
CAS 738: AlgebraicMethods
in SoftwareEngineering
and ComputerScience
Dr. R. Khedri
Outline
Introduction
Binary relations
Proper relation algebras
Square proper R.A.
Relation algebras
Atom structures ofRAs
Examples of relationalgebra
(Slide 6 of 38)Relation Algebras Binary relations
We considered fields of unary relations (or fields ofsets), and saw that the corresponding algebra wasBoolean algebra
We start our investigation of binary relations bydefining a field of binary relations
A binary relation on a set B is a subset of B × BB is said to be the base set
on B we can define a (proper) relation algebra
CAS 738: AlgebraicMethods
in SoftwareEngineering
and ComputerScience
Dr. R. Khedri
Outline
Introduction
Binary relations
Proper relation algebras
Square proper R.A.
Relation algebras
Atom structures ofRAs
Examples of relationalgebra
(Slide 7 of 38)Relation Algebras Binary relations Properrelation algebras
Definition
Let B be a set. A proper relation algebra (PRA) with baseset B is an algebra of the form S = (S , ∅, U,∪, \, IB ,` , ;),where S is a non-empty set of binary relations over B, andthe following hold:
1 (S , ∅, U,∪, \) forms a field of sets
2 IBdef= (b, b) | b ∈ B ∈ S (the identity over B)
3 S is closed under taking converses
s ∈ S =⇒ s` ∈ S , where s` = (c , b) | (b, c) ∈ s.
4 S is closed under composition of binary relations:r , s ∈ S =⇒ r ; s ∈ S , where
r ; s = (b, c) | ∃(d |: (b, d) ∈ r ∧ (d , c) ∈ s ).
CAS 738: AlgebraicMethods
in SoftwareEngineering
and ComputerScience
Dr. R. Khedri
Outline
Introduction
Binary relations
Proper relation algebras
Square proper R.A.
Relation algebras
Atom structures ofRAs
Examples of relationalgebra
(Slide 8 of 38)Relation Algebras Binary relations Properrelation algebras
The unit U does not need to be B × B
Example
B = 0, 1, 2S contains only the following relations:
∅ (0, 0), (1, 1) (2, 2)(0, 1), (1, 0) (0, 0), (1, 1), (2, 2) = IB (0, 0), (1, 1), (0, 1), (1, 0)(2, 2), (0, 1), (1, 0) (0, 0), (1, 1), (2, 2), (0, 1), (1, 0) = U
CAS 738: AlgebraicMethods
in SoftwareEngineering
and ComputerScience
Dr. R. Khedri
Outline
Introduction
Binary relations
Proper relation algebras
Square proper R.A.
Relation algebras
Atom structures ofRAs
Examples of relationalgebra
(Slide 8 of 38)Relation Algebras Binary relations Properrelation algebras
The unit U does not need to be B × B
Example
B = 0, 1, 2S contains only the following relations:
∅ (0, 0), (1, 1) (2, 2)(0, 1), (1, 0) (0, 0), (1, 1), (2, 2) = IB (0, 0), (1, 1), (0, 1), (1, 0)(2, 2), (0, 1), (1, 0) (0, 0), (1, 1), (2, 2), (0, 1), (1, 0) = U
CAS 738: AlgebraicMethods
in SoftwareEngineering
and ComputerScience
Dr. R. Khedri
Outline
Introduction
Binary relations
Proper relation algebras
Square proper R.A.
Relation algebras
Atom structures ofRAs
Examples of relationalgebra
(Slide 9 of 38)Relation Algebras Binary relations Squareproper relation algebras
To remember
the complement \ in a proper relation algebra is alwaystaken relative to the unit UU of a proper relation algebra with base B need not beB × B
In terms of the previous definition, (S , ∅, U,∪, \) isNOT necessarily a subalgebra of the field of sets(P(B × B), ∅,B × B,∪, \)
It is a subalgebra of the field of sets (P(U), ∅, U,∪, \)
CAS 738: AlgebraicMethods
in SoftwareEngineering
and ComputerScience
Dr. R. Khedri
Outline
Introduction
Binary relations
Proper relation algebras
Square proper R.A.
Relation algebras
Atom structures ofRAs
Examples of relationalgebra
(Slide 10 of 38)Relation Algebras Binary relations Squareproper relation algebras
Several reasons for not insisting that U = B × B
relativising the operations to the unit, leads to otherclasses of algebras which are in many waysbetter-behaved
to ensure that the class of algebras isomorphic toproper relation algebras is a variety (has nice properties)
Nonetheless, those proper relation algebras whereU = B × B are very important
CAS 738: AlgebraicMethods
in SoftwareEngineering
and ComputerScience
Dr. R. Khedri
Outline
Introduction
Binary relations
Proper relation algebras
Square proper R.A.
Relation algebras
Atom structures ofRAs
Examples of relationalgebra
(Slide 11 of 38)Relation Algebras Binary relations Squareproper relation algebras
Definition (Proper relation algebra)
A proper relation algebra S with base B is said to be squareif its unit U is equal to B × B.
An arbitrary proper relation algebra S need not besquare
But its unit will always be an equivalence relation
Lemma
Let S = (S , ∅, U,∪, \, IB ,` , ;) be a proper relation algebrawith base B. Then the unit U is an equivalence relationover B.
Proof: . . .
CAS 738: AlgebraicMethods
in SoftwareEngineering
and ComputerScience
Dr. R. Khedri
Outline
Introduction
Binary relations
Proper relation algebras
Square proper R.A.
Relation algebras
Atom structures ofRAs
Examples of relationalgebra
(Slide 12 of 38)Relation Algebras Binary relations Squareproper relation algebras
New construction: Restricted proper relation algebra (RPA)
Definition (restricted PRA)
Let S = (S , ∅, U,∪, \, IB ,` , ;) be a proper relation algebrawith base B. Let E ⊆ B be an equivalence class of U.Define the restricted proper relation algebra
S↓E
def= (S↓
E, ∅,E × E ,∪, \
E, IE ,` , ;), where
S↓E
def= s ∩ E × E | s ∈ S
\E(s) = (E × E ) \ s, for s ∈ S .
As example (S↓E= s | s ⊂ IB, for a given B)
CAS 738: AlgebraicMethods
in SoftwareEngineering
and ComputerScience
Dr. R. Khedri
Outline
Introduction
Binary relations
Proper relation algebras
Square proper R.A.
Relation algebras
Atom structures ofRAs
Examples of relationalgebra
(Slide 13 of 38)Relation Algebras Binary relations Squareproper relation algebras
Another new construction: Proper relation algebra withbase B = ∪(i | i ∈ I : Bi )
Definition
Let Si = (Si , ∅, Ui ,∪, \i , IBi,` , ;) be a proper relation
algebra with base Bi for each i ∈ I (some index set I ), andsuppose that if i 6= j in I then Bi ∩ Bj = ∅. We define
⊗i∈ISi = (S , ∅, U,∪, \, IB ,` , ;)
to be the proper relation algebra with baseB = ∪(i | i ∈ I : Bi ), where:
S = ∪(i | i ∈ I : si ) | si ∈ Si for each i ∈ I(we are including arbitrary unions of the relations)
U = ∪(i | i ∈ I : Ui )
CAS 738: AlgebraicMethods
in SoftwareEngineering
and ComputerScience
Dr. R. Khedri
Outline
Introduction
Binary relations
Proper relation algebras
Square proper R.A.
Relation algebras
Atom structures ofRAs
Examples of relationalgebra
(Slide 14 of 38)Relation Algebras Binary relations Squareproper relation algebras
Lemma
Let S = (S , ∅, U,∪, \, IB ,` , ;) be a proper relation algebrawith base B, and we write E for the set B/U of equivalenceclasses of the unit U. Then
S is a subalgebra of ⊗E∈E(S↓E), and
for each E ∈ E , there is a homomorphism mapping Sonto S↓
E.
Proof: . . .
CAS 738: AlgebraicMethods
in SoftwareEngineering
and ComputerScience
Dr. R. Khedri
Outline
Introduction
Binary relations
Relation algebras
Definition of RAs
Peircean law
Atom structures ofRAs
Examples of relationalgebra
(Slide 15 of 38)Relation Algebras Relation algebras
Having defined a proper relation algebra, we now seekan algebraic counterpart
We present certain axioms defining abstract relationalgebras
We try to show, among other things, the relationshipof relation algebras to Boolean algebras
CAS 738: AlgebraicMethods
in SoftwareEngineering
and ComputerScience
Dr. R. Khedri
Outline
Introduction
Binary relations
Relation algebras
Definition of RAs
Peircean law
Atom structures ofRAs
Examples of relationalgebra
(Slide 16 of 38)Relation Algebras Relation algebrasDefinition of relation algebras
Definition
A relation algebra is an algebra A = (A, 0, 1,+,−, 1′,` , ;)satisfying the following axioms, for all a, b, c ∈ A:
R1 The equations defining a Boolean algebra
R2 a ; (b ; c) = (a ; b) ; c
R3 (a + b) ; c = a ; c + b ; c
R4 a ; 1′ = a
R5 a`` = a
R6 (a + b)` = a` + b`
R7 (a ; b)` = b` ; a`
R8 a` ; (−(a ; b)) ≤ −b
We use RA to denote the class of all relation algebras
CAS 738: AlgebraicMethods
in SoftwareEngineering
and ComputerScience
Dr. R. Khedri
Outline
Introduction
Binary relations
Relation algebras
Definition of RAs
Peircean law
Atom structures ofRAs
Examples of relationalgebra
(Slide 17 of 38)Relation Algebras Relation algebrasDefinition of relation algebras
1′ is the identity
` is the conversion
; is the composition
Binding conventions:` is tighter than −all unary operations are tighter than binary ones; tighter · tighter +
The operations in descending order of priority are` ,−, ;, ·,+
CAS 738: AlgebraicMethods
in SoftwareEngineering
and ComputerScience
Dr. R. Khedri
Outline
Introduction
Binary relations
Relation algebras
Definition of RAs
Peircean law
Atom structures ofRAs
Examples of relationalgebra
(Slide 18 of 38)Relation Algebras Relation algebrasDefinition of relation algebras
What is the language that we use to describe relationalgebras?
Definition
LRA is the functional signature consisting of constants0, 1, 1′, unary function symbols −,` , and the binaryfunction symbols +, ;.
LRA-Structures are often called relation-type algebras
The interpretations of the symbols 0, 1, 1′ in anLRA-structure A are denoted by 0A, 1A, 1′A
Recall LRA-term, LRA-formulas . . .
The axioms for relation algebras are LRA-equations
CAS 738: AlgebraicMethods
in SoftwareEngineering
and ComputerScience
Dr. R. Khedri
Outline
Introduction
Binary relations
Relation algebras
Definition of RAs
Peircean law
Atom structures ofRAs
Examples of relationalgebra
(Slide 19 of 38)Relation Algebras Relation algebrasDefinition of relation algebras
` is bijective` is order preserving
1` = 1 and 0` = 0
(−a)` = −a`
Lemma
In any relation algebra A, ` is an automorphism of theBoolean reduct bool(A).
Proof: We must show that ` is bijective and that 1` = 1,0` = 0, and∀(a, b | a, b ∈ A : (a+B)` = a` +b` ∧ (−a)` = −a` ).
CAS 738: AlgebraicMethods
in SoftwareEngineering
and ComputerScience
Dr. R. Khedri
Outline
Introduction
Binary relations
Relation algebras
Definition of RAs
Peircean law
Atom structures ofRAs
Examples of relationalgebra
(Slide 20 of 38)Relation Algebras Relation algebrasPeircean law
How to understand[a` ; (−(a ; b)) ≤ −b
](i.e., R8) in a
more intuitive and useful form?
R8 is known as the Peircean law or De Morgan’sTheorem K
Definition
The Peircean law is the LRA-sentence
∀(x , y , z |: (x ; y) · z` = 0 ⇐⇒ (y ; z) · x` = 0 ) (PL)
We can continue and add a third equivalence,(z ; x) · y` = 0
Lemma
Axioms R1, R3, R5, R6, R7 |= (axiom R8 ⇐⇒ PL),
CAS 738: AlgebraicMethods
in SoftwareEngineering
and ComputerScience
Dr. R. Khedri
Outline
Introduction
Binary relations
Relation algebras
Definition of RAs
Peircean law
Atom structures ofRAs
Examples of relationalgebra
(Slide 21 of 38)Relation Algebras Relation algebrasPeircean law
CAS 738: AlgebraicMethods
in SoftwareEngineering
and ComputerScience
Dr. R. Khedri
Outline
Introduction
Binary relations
Relation algebras
Atom structures ofRAs
BAO
Atom structures of RAs
Consistent andforbidden triples ofatoms
Representations of RAs
Examples of relationalgebra
(Slide 22 of 38)
Relation AlgebrasAtom structures of relation algebrasBoolean algebras with operator
Boolean algebras with operators
We now generalise the notion of Boolean algebras byadding extra functions (or operators)
What results is called a Boolean algebra withoperators, or BAO
This more general setting was proposed by Jonsson andTarski in 1951,
BAOs are helpful when we come to consider algebras ofbinary and higher-order relations
CAS 738: AlgebraicMethods
in SoftwareEngineering
and ComputerScience
Dr. R. Khedri
Outline
Introduction
Binary relations
Relation algebras
Atom structures ofRAs
BAO
Atom structures of RAs
Consistent andforbidden triples ofatoms
Representations of RAs
Examples of relationalgebra
(Slide 23 of 38)
Relation AlgebrasAtom structures of relation algebrasBoolean algebras with operatorBoolean algebras with operators
Definition (Operator)
Let B = 〈B, 0, 1,+,−〉 be a boolean algebra. An operatorΩ with arity or rank rk(Ω) = n < ω onB is a functionΩ : Bn −→ B such that:
1 Ω is normal:
∀(b0, · · · , bn−1 | b0, · · · , bn−1 ∈ B ∧ i < n :
bi = 0 =⇒ Ω(b0, · · · , bn−1) = 0 )
2 Ω is additive:
∀(b0, · · · , bn−1, b, b′ | b0, · · · , bn−1, b, b′ ∈ B ∧ i < n :
Ω (b0, · · · , bi−1, (b + b′), · · · , bn−1)
= Ω (b0, · · · , bi−1, b, · · · , bn−1)
+Ω (b0, · · · , bi−1, b′, · · · , bn−1) )
We may call Ω an n-ary operator.
CAS 738: AlgebraicMethods
in SoftwareEngineering
and ComputerScience
Dr. R. Khedri
Outline
Introduction
Binary relations
Relation algebras
Atom structures ofRAs
BAO
Atom structures of RAs
Consistent andforbidden triples ofatoms
Representations of RAs
Examples of relationalgebra
(Slide 24 of 38)
Relation AlgebrasAtom structures of relation algebrasBoolean algebras with operator
Definition (BAO)
A Boolean algebra with operators (BAO) is an algebra〈B, 0, 1,+,−,Ωλ : λ ∈ Λ〉, where B = 〈B, 0, 1,+,−〉 is aboolean algebra, Λ is a set (perhaps uncountable), andΩλ(λ ∈ Λ) are operators on B.
Definition
Let B = 〈B, 0, 1,+,−〉 be a boolean algebra, and f an n-aryfunction on B. f is said to be completely additive if itsatisfies,
∀(b0, · · · , bn−1,X | b0, · · · , bn−1 ∈ B∧ i < n ∧ X ⊆ B ∧ ΣX exists in B :
f (b0, · · · ,ΣX , · · · bn−1) = Σx∈X f (b0, · · · , x , · · · bn−1) )
for all a, b ∈ B. So f is normal and additive, and hence anoperator on B.
CAS 738: AlgebraicMethods
in SoftwareEngineering
and ComputerScience
Dr. R. Khedri
Outline
Introduction
Binary relations
Relation algebras
Atom structures ofRAs
BAO
Atom structures of RAs
Consistent andforbidden triples ofatoms
Representations of RAs
Examples of relationalgebra
(Slide 25 of 38)
Relation AlgebrasAtom structures of relation algebrasBoolean algebras with operator
A BAO is said to be completely additive if each of itsoperators is completely additive
A class K of BAOs is completely additive if each BAOin K is completely additive
Lemma
Any normal completely additive n-ary function f on aBoolean algebra B is an operator on B.
Proof: . . .
CAS 738: AlgebraicMethods
in SoftwareEngineering
and ComputerScience
Dr. R. Khedri
Outline
Introduction
Binary relations
Relation algebras
Atom structures ofRAs
BAO
Atom structures of RAs
Consistent andforbidden triples ofatoms
Representations of RAs
Examples of relationalgebra
(Slide 26 of 38)
Relation AlgebrasAtom structures of relation algebrasBoolean algebras with operator
Definition
Let L be a functional signature containing the signatureLBA of boolean algebras. We regard constants of L asnullary function symbols.
1 For each n-ary operator symbol Ω ∈ L \ LBA, introducean (n + 1)-ary relation symbol RΩ. Let La be therelational signature RΩ : Ω ∈ L \ LBA.
2 Let B be an atomic BAO of signature L. The atomstructure of B is defined to be the La-structure withdomain the set At(B) of atoms of B and with relationsdefined by:At(B) |= RΩ(a0, ..., an−1, b)
⇐⇒ B |= (b ≤ Ω(a0, ..., an−1))for each n-ary Ω ∈ L \ LBA and atoms a0, ..., an−1, b ofB.
CAS 738: AlgebraicMethods
in SoftwareEngineering
and ComputerScience
Dr. R. Khedri
Outline
Introduction
Binary relations
Relation algebras
Atom structures ofRAs
BAO
Atom structures of RAs
Consistent andforbidden triples ofatoms
Representations of RAs
Examples of relationalgebra
(Slide 27 of 38)
Relation AlgebrasAtom structures of relation algebrasBoolean algebras with operator
For a completely additive operator Ω over an atomicBAO, we can calculate Ω if we know how it is definedon the atoms
The behaviour of the operators on the atoms can bedefined by specifying the atom structure
This is one reason why atoms are so important
The following definitions and propositions are just a(long-winded) way of saying that you can work equallywell with an atomic, completely additive BAO
It’s easier to work with the atom structure (working atthe frame level)
They also allow us to construct various new BAOs fromold ones, by building atom structures
CAS 738: AlgebraicMethods
in SoftwareEngineering
and ComputerScience
Dr. R. Khedri
Outline
Introduction
Binary relations
Relation algebras
Atom structures ofRAs
BAO
Atom structures of RAs
Consistent andforbidden triples ofatoms
Representations of RAs
Examples of relationalgebra
(Slide 28 of 38)
Relation AlgebrasAtom structures of relation algebrasAtom structures of RAs
Relation algebras are (completely additive) BAOs
We can apply our definitions and results about atomstructures
The most convenient way to specify an atomic relationalgebra is by giving the atom structure
This means listingthe atoms underneath the identity,the pairs of atoms (a, b) such that b ≤ a`
the triples of atoms (a, b, c) such that c ≤ a ; b
We end with a structure 〈X ,R1′ ,R` ,R;〉
CAS 738: AlgebraicMethods
in SoftwareEngineering
and ComputerScience
Dr. R. Khedri
Outline
Introduction
Binary relations
Relation algebras
Atom structures ofRAs
BAO
Atom structures of RAs
Consistent andforbidden triples ofatoms
Representations of RAs
Examples of relationalgebra
(Slide 29 of 38)
Relation AlgebrasAtom structures of relation algebrasAtom structures of RAs
Because we are dealing with relation algebras, we cansimplify the format of atom structures slightly
We will replace the binary relation symbol R` by theunary function symbol ` , and read R` (x , y) as x` = y
Definition
The atom structure, written At(A), of an atomic relationalgebra A is the structure
〈S , a ∈ S | a ≤ 1′, (a 7→ a`)a∈S , (a, b, c) ∈ S | c ≤ a ; b〉,
where S is the set of atoms ofA. The signature of thisstructure consists of a unary relation symbol R1′ , a unaryfunction symbol ` , and a ternary relation symbol R;.
CAS 738: AlgebraicMethods
in SoftwareEngineering
and ComputerScience
Dr. R. Khedri
Outline
Introduction
Binary relations
Relation algebras
Atom structures ofRAs
BAO
Atom structures of RAs
Consistent andforbidden triples ofatoms
Representations of RAs
Examples of relationalgebra
(Slide 30 of 38)
Relation AlgebrasAtom structures of relation algebrasAtom structures of RAs
Working in the other direction:
Given any structure A in the signature R1′ ,` ,R;, wecan define a conventional La
RA-structure A ′ by letting
(R` )A ′ = (a, a`) | a ∈ A
From A ′, we can define an LRA-BAO
Problem: it may fail to be a relation algebra (e.g.,a`` 6= a)
So we want to find axioms for those structures in thesignature R1′ ,` ,R; that are the atom structures ofrelation algebras
CAS 738: AlgebraicMethods
in SoftwareEngineering
and ComputerScience
Dr. R. Khedri
Outline
Introduction
Binary relations
Relation algebras
Atom structures ofRAs
BAO
Atom structures of RAs
Consistent andforbidden triples ofatoms
Representations of RAs
Examples of relationalgebra
(Slide 31 of 38)
Relation AlgebrasAtom structures of relation algebrasAtom structures of RAs
Definition
A structure S for the signature R1′ ,` ,R; is called arelation algebra atom structure if it satisfies:
1 identity: ∀(x , y | x , y ∈ S : x = y ) ⇐⇒ ∃(e |e ∈ S : R1′(e) ∧ R;(x , e, y) )
2 ∀(x , y , z | x , y , z ∈ S : R;(x , y , z)=⇒ R;(x`, z , y) ∧ R;(y`, x`, z`) )
3 Associativity:
∀(x , y , z , t | x , y , z , t ∈ S :
∃(u | u ∈ S : R;(x , y , u) ∧ R;(u, z , t) )
⇐⇒ ∃(v | v ∈ S : R;(y , z , v) ∧ R;(x , v , t) ) ).
CAS 738: AlgebraicMethods
in SoftwareEngineering
and ComputerScience
Dr. R. Khedri
Outline
Introduction
Binary relations
Relation algebras
Atom structures ofRAs
BAO
Atom structures of RAs
Consistent andforbidden triples ofatoms
Representations of RAs
Examples of relationalgebra
(Slide 32 of 38)
Relation AlgebrasAtom structures of relation algebrasConsistent and forbidden triples of atoms
In practice, we will often specify the compositionrelation R; of a relation algebra atom structure bylisting its consistent triples
Aternatively, we can list the inconsistent or forbiddentriples instead
CAS 738: AlgebraicMethods
in SoftwareEngineering
and ComputerScience
Dr. R. Khedri
Outline
Introduction
Binary relations
Relation algebras
Atom structures ofRAs
BAO
Atom structures of RAs
Consistent andforbidden triples ofatoms
Representations of RAs
Examples of relationalgebra
(Slide 33 of 38)
Relation AlgebrasAtom structures of relation algebrasConsistent and forbidden triples of atoms
Definition
Let A be an atomic relation algebra, and S a relationalgebra atom structure.
A triple (a, b, c) of atoms of A is said to be consistentif c` ≤ a ; b.
A triple (a, b, c) of atoms of S is said to be consistentif R;(a, b, c`)
A triple (a, b, c) of atoms of A or elements of S is saidto be inconsistent or forbidden, if it is not consistent(i.e., a ; b · c` = 0 or ¬R;(a, b, c`), resp.)
The six Peircean transforms of a triple (a, b, c) ofatoms are
(a, b, c), (b, c , a), (c , a, b), (a`, c`, b`), (b`, a`, c`), (c`, b`, a`)
CAS 738: AlgebraicMethods
in SoftwareEngineering
and ComputerScience
Dr. R. Khedri
Outline
Introduction
Binary relations
Relation algebras
Atom structures ofRAs
BAO
Atom structures of RAs
Consistent andforbidden triples ofatoms
Representations of RAs
Examples of relationalgebra
(Slide 34 of 38)
Relation AlgebrasAtom structures of relation algebrasRepresentations of relation algebras
Definition
Let A be a relation algebra.
A representation of A is an isomorphism from A to aproper relation algebra
A is said to be representable if it has a representation
A representation h : A −→ P is said to be square if Pis a square proper relation algebra
CAS 738: AlgebraicMethods
in SoftwareEngineering
and ComputerScience
Dr. R. Khedri
Outline
Introduction
Binary relations
Relation algebras
Atom structures ofRAs
Examples of relationalgebra
(Slide 35 of 38)Relation Algebras Examples of relation algebra
The smallest relation algebra
This is of course the one-element (degenerate) relationalgebra
It is unique up to isomorphism
A representation with base 0 is obtained by mapping itsunique element to 0
The algebra is isomorphic to F(∅)
CAS 738: AlgebraicMethods
in SoftwareEngineering
and ComputerScience
Dr. R. Khedri
Outline
Introduction
Binary relations
Relation algebras
Atom structures ofRAs
Examples of relationalgebra
(Slide 36 of 38)Relation Algebras Examples of relation algebra
The smallest non-degenerate relation algebra
It has one atom (i.e., 1) and two elements, 0, 1
It is based on the two element boolean algebra with thefollowing operators:
1′ = 1both elements are self-converseand composition is defined by 1 ; 1 = 1, 0 ; a = a ; 0 = 0for all a ∈ 0, 1
For a representation, take a set X with exactly onepoint x
Let h(0) = ∅ and h(1) = (x , x)The algebra is isomorphic to F(x)
CAS 738: AlgebraicMethods
in SoftwareEngineering
and ComputerScience
Dr. R. Khedri
Outline
Introduction
Binary relations
Relation algebras
Atom structures ofRAs
Examples of relationalgebra
(Slide 37 of 38)Relation Algebras Examples of relation algebra
Two-atom algebra
This relation algebra has two atoms 1′ and # (therefore has four elements)
Both atoms are self-converse
Composition is defined by; 1′ #
1′ 1′ #
# # 1 (or 1′)
Alternatively, the forbidden triples of atoms are
(1′, 1′,#), (1′,#, 1′), (#, 1′, 1′), (#,#,#)
CAS 738: AlgebraicMethods
in SoftwareEngineering
and ComputerScience
Dr. R. Khedri
Outline
Introduction
Binary relations
Relation algebras
Atom structures ofRAs
Examples of relationalgebra
(Slide 38 of 38)
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