cas 738 relation algebra and kleene algebra and their ...€¦ · representations of boolean...

CAS 738 RelationAlgebra

and Kleene Algebraand their

Applications

Dr. R. Khedri

Outline

Introduction

Definition andexamples

Basic Definitions

Atoms

Dense sets

Ideals, filters, andultrafilters

Representations ofBoolean algebras

Complete BooleanAlgebra

(Slide 1 of 32)

CAS 738 Relation Algebraand Kleene Algebra

and their Applications

Dr. Ridha Khedri

Department of Computing and Software, McMaster UniversityCanada L8S 4L7, Hamilton, Ontario



Applications

Dr. R. Khedri

Outline

Introduction


Basic Definitions

Atoms

Dense sets




(Slide 2 of 32)

1 Introduction

2 Definition and examples

3 Basic DefinitionsExample of BAs

4 Atoms

5 Dense sets

6 Ideals, filters, and ultrafilters

7 Representations of Boolean algebras

8 Complete Boolean Algebra



Applications

Dr. R. Khedri

Outline

Introduction


Basic Definitions

Atoms

Dense sets




(Slide 3 of 32)Boolean Algebras Introduction

In 1847 George Boole defined a calculus of propositionsESSAY: The Mathematical Analysis of Logic

Later this calculus developed into the subject we nowcall Boolean algebra

It found successful application throughout mathematicsand computing

Relation algebras are motivationally and technicallybased on Boolean algebras

In this set of slides, we outline the parts of the theoryof Boolean algebras



Applications

Dr. R. Khedri

Outline

Introduction


Basic Definitions

Atoms

Dense sets




(Slide 4 of 32)Boolean Algebras Definition and examples

The best way of thinking of Boolean algebra is to start withthe notion of a field of sets

Definition (LBA)

Let LBA be the functional signature with constants 0 and 1,a binary function symbol +, and a unary function symbol .



Applications

Dr. R. Khedri

Outline

Introduction


Basic Definitions

Atoms

Dense sets





Definition (LBA-algebra)

Let X be any set (the “base set”). A field of sets over thebase X is an LBA-algebra F = (F , ∅,X ,∪, \), where F is anon-empty set of subsets of X (so, ∅ ⊂ F ⊆ P(X )), suchthat if S ,T ∈ F then (S ∪ T ) ∈ F and (X \ S) ∈ F .

0 and 1 are interpreted in F as ∅ and X

+ is interpreted as ∪is interpreted as the unary function S −→ X \ S

Example

X = {1, 2, 3}F = {∅,X , {1}, {2, 3}}In this case, we have ∅ ⊂ F ⊂ P(X )



Applications

Dr. R. Khedri

Outline

Introduction


Basic Definitions

Atoms

Dense sets





So, a field of sets is closed under ∪ and \It follows that F contains ∅, which is X

F is an LBA-algebra

Note that it is also closed under finite intersections

The word “field” is used to indicate this closedness

A field of sets is a rather important example of aBoolean algebra

The idea with Boolean algebras is that we treat theset-theoretic operations ∪ and \ as abstract operations+ and



Applications

Dr. R. Khedri

Outline

Introduction


Basic Definitions

Atoms

Dense sets





A Boolean algebra does not have to consist of sets withthe operations of union and complement

The elements of the domain of a Boolean algebra arearbitrary

The operations and + become formal operations thatsatisfy certain axioms

So, we are dealing with models of a certain LBA-theory

In models B, we should be writing 0B, 1B,+B, B

when no confusion is likely, we simply canwrite 0, 1,+,



Applications

Dr. R. Khedri

Outline

Introduction


Basic Definitions

Atoms

Dense sets





Definition (Boolean Algebra)

Let B = 〈B, 0, 1,+,−〉 be anLBA-structure. B is a Booleanalgebra if it satisfies the following equations, for alla, b, c ∈ B:

+ is associative, commutative and idempotent1 (a + b) + c = a + (b + c)2 a + b = b + a3 a + a = a

complement4 −(−b) = b5 b + (−b) = 16 −1 = 0

connections of · and +7 a · (b + c) = a · b + a · c , where a · b abbreviates−(−a +−b)

zero8 0 + a = a



Applications

Dr. R. Khedri

Outline

Introduction


Basic Definitions

Atoms

Dense sets





Standard abbreviations:

We let a ≤ b abbreviate the equation a + b = b

We let a < b abbreviate the formula a ≤ b ∧ a 6= b

We sometimes use − as a binary operator, lettinga− b = a · (−b)

≤ defines a partial order (reflexive, transitive, andantisymmetric) on B

a ≤ b ⇐⇒ a · b = a ⇐⇒ a · −b = 0

In a field of sets F , we have

∀(a, b | a, b ∈ F : (a ≤ b) ⇐⇒ (a+b = a∪b = b) ⇐⇒ (a ⊆ b) )

Therefore, ≤ is set inclusion



Applications

Dr. R. Khedri

Outline

Introduction


Basic Definitions

Atoms

Dense sets





Definition (Degenerate B.A.)

A Boolean algebra is said to be degenerate if it has exactlyone element. Up to isomorphism, there is a uniquedegenerate Boolean algebra.

Definition (Boolean homomorphism)

A Boolean homomorphism is a homomorphism from aBoolean algebra A to another B.

Because we are dealing with algebras, a Booleanhomomorphism is an isomorphism iff it is a bijection



Applications

Dr. R. Khedri

Outline

Introduction


Basic Definitions

Example of BAs

Atoms

Dense sets




(Slide 11 of 32)Boolean Algebras Basic Definitions Example ofBAs

Example 1

Let L be any logic including the propositionalconnectives ∨ and ¬ and with a standard notion ofproof |=Let form(L) be the set of all L-formulasTake a set Γ that is a set of consistent L-formulas

Can we construct a Boolean Algebra on it?

We defines an equivalence relation ∼ on form(L) where

φ ∼ ψ ⇐⇒ Γ |= (φ ⇐⇒ ψ)

For every φ, ψ ∈ form(L),φ/∼ denotes the ∼-equivalence class of φφ/∼ + ψ/∼ = (φ ∨ ψ)/∼−(φ/∼) = (¬φ)/∼For typical |=, + and - are well-defined⟨

form(L)/∼,⊥/∼,>/∼,+,−⟩

is a Boolean algebra



Applications

Dr. R. Khedri

Outline

Introduction


Basic Definitions

Example of BAs

Atoms

Dense sets




(Slide 12 of 32)Boolean Algebras Basic Definitions Example ofBAs

Example 2

Let L be any logic including the propositionalconnectives ∨ and ¬ and with a standard notion ofproof |=Let K be a class of L-structures

Another Boolean Algebra on it?

We defines an equivalence relation ' on form(L) where

φ ' ψ ⇐⇒ K |= (φ ⇐⇒ ψ)

For every φ, ψ ∈ form(L),φ/' denotes the '-equivalence class of φφ/' + ψ/' = (φ ∨ ψ)/'−(φ/') = (¬φ)/'For typical |=, + and - are well-defined⟨

form(L)/',⊥/',>/',+,−⟩

is a Boolean algebra



Applications

Dr. R. Khedri

Outline

Introduction


Basic Definitions

Atoms

Dense sets




(Slide 13 of 32)Boolean Algebras Atoms

Many properties of Boolean algebras and relatedalgebras reduce to properties of the ”simplest”elements of their structures

These elements are called atoms.

The notion of an atom is therefore quite relevant todefine and study

Definition (Atom)

Let B be a Boolean algebra. An atom of B is a minimalnon-zero element a ∈ B

∀(a′ | a′ ∈ B : a′ < a ⇐⇒ a′ = 0 )

We write At(B) for the set of all atoms of B.



Applications

Dr. R. Khedri

Outline

Introduction


Basic Definitions

Atoms

Dense sets





Definition (Atomic B.A.)

Let B be a Boolean algebra. B is said to be atomic if

∀(b | b ∈ B ∧ b 6= 0 : ∃(a | a ∈ At(B) : a ≤ b ) ).

Example

A field of sets of the form F = (P(X ), ∅,X ,∪, \) is atomicThe atoms are the singleton sets

Definition (Complete)

A Boolean algebra B is said to be complete if∀(S | S ⊆ B : ΣS and ΠS exist )

ΣS is the supremum or joint or sum of S

ΠS is the infimum or meet or product of S



Applications

Dr. R. Khedri

Outline

Introduction


Basic Definitions

Atoms

Dense sets





Results on finite Boolean algebras

Any finite Boolean algebra is atomic

Two complete and atomic Boolean algebras with thesame number of atoms are isomorphic

Every bijection between the sets of atoms extends toan isomorphism betweens the algebras

Any finite Boolean algebra A is isomorphic to the fieldF = (P(X ), ∅,X ,∪, \), where |X | = n. In fact n is thenumber of atoms.

Two finite Boolean algebras with the same cardinalityare isomorphic

There exist infinite atomic Boolean algebras



Applications

Dr. R. Khedri

Outline

Introduction


Basic Definitions

Atoms

Dense sets





Definition

A Boolean algebra B is said to be atomless if it isnon-degenerate and has no atoms.

Example of atomless Boolean Algebra

Let X ⊆ RWe define MX = {I : interval | I ⊆ X ∧ ∀(J :interval | I ⊆ J ⊆ X : J = I )}, which is called theset of maximal subintervals of X

X is an open set⇐⇒ ∀(I | I ∈ MX : I is an open interval )

X is regular set⇐⇒ ∀(I , I | I , J ∈ MX : I , J are apart ) (i.e., I andJ do not overlap)



Applications

Dr. R. Khedri

Outline

Introduction


Basic Definitions

Atoms

Dense sets





Example of atomless Boolean Algebra (Continued ....)

Let S = {X | X ⊆ R ∧ X is regular and open}S is said to be the set of all solids in R

S = (S , ∅,R,∩,∼), where

∼ X = (R− X )− {Y ∈ R | ∃(I | I ∈ M(R−X ) :

Y is lower bound of I or the upper bound of I )}X ∪ Y =∼∼ (X ∪ Y )

S is a Boolean algebra

S is a atomless



Applications

Dr. R. Khedri

Outline

Introduction


Basic Definitions

Atoms

Dense sets




(Slide 18 of 32)Boolean Algebras Dense sets

The atoms of an atomic Boolean algebra form an exampleof a dense set

Definition (Dense subset of B)

A dense subset D of B is one such that if b ∈ B \ {0} thenthere exists d ∈ D \ {0} with d ≤ b.

Definition (Dense subalgebra of B)

A subalgebra C, with a support set C , of B is said to be adense subalgebra of B if C is dense in B.



Applications

Dr. R. Khedri

Outline

Introduction


Basic Definitions

Atoms

Dense sets




(Slide 19 of 32)Boolean Algebras Ideals, filters, and ultrafilters

Filters and ideals play an important role in severalmathematical disciplines

There is a duality between filters and ideals

Mathematicians and theoretical computer scientists canbe divided into two camps according to whether theyprefer to work with ideals or filters

In the following we mostly use filters

Definition (Ideal)

An ideal over B is a non-empty subset I of the domain ofcalB (written I ⊆ B) such that

a) I is closed downwards:∀(s | s ∈ I : t ≤ s =⇒ t ∈ I ),

b) I is closed under +: ∀(s, t | s, t ∈ I : s + t ∈ I ).



Applications

Dr. R. Khedri

Outline

Introduction


Basic Definitions

Atoms

Dense sets





The dual of an ideal is the filter

Definition (Filter)

A filter over B is a non-empty subset F of the domain ofcalB (written F ⊆ B) such that

a) F is closed upwards: ∀(s | s ∈ F : t ≥ s =⇒ t ∈ F ),

b) I is closed under ·: ∀(s, t | s, t ∈ F : s · t ∈ F ).

Clearly, the support set of B forms a filter on BAny other filter is said to be proper

{1} is a trivial filter of BAny other filter is said to be non-trivial

The same goes for ideals



Applications

Dr. R. Khedri

Outline

Introduction


Basic Definitions

Atoms

Dense sets





For any b ∈ B, let I (b) = {c ∈ B | b ≥ c} andF (b) = {c ∈ B | b ≤ c}I (b) and F (b) are called, respectively, the principalideal and filter generated by b

Any filter or ideal not of this form is said to benon-principal

A maximal ideal I is a proper ideal that is not strictlycontained in any other proper ideal

An ultrafilter F is a proper filter not strictly containedin any other proper filter

A filter is proper iff it does not contain 0

An equivalent definition of an ultrafilter is a filter Fsuch that∀(b | b ∈ B : exactly one of b and − b ∈ F )



Applications

Dr. R. Khedri

Outline

Introduction


Basic Definitions

Atoms

Dense sets





Theorem (Boolean prime ideal theorem, ’BPI’)

Let B be a Boolean algebra, and let S ⊆ B be a subset suchthat s0 · s1 . . . · sn−1 6= 0 for any n < ω and anys0, · · · , sn−1 ∈ S. Then there exists an ultrafilter of Bcontaining S.

When conditions of the BPI are satisfied, we say that Shas the finite intersection property

Any proper filter has the finite intersection property

A degenerate Boolean algebra has no ultrafilters

A degenerate Boolean algebra has no subset with thefinite intersection property, so BPI holds vacuously inthis case



Applications

Dr. R. Khedri

Outline

Introduction


Basic Definitions

Atoms

Dense sets





Note on set theory

The Boolean prime ideal (BPI) theorem cannot beproved without some form of the axiom of choice, suchas Zorn’s lemma

BPI is actually weaker than the full axiom of choice

Modulo ZF, BPI is equivalent to the compactnesstheorem for first-order logic, and to the statement thatany set can be linearly ordered

The axiom of choice is equivalent to the statementthat any set can be well-ordered



Applications

Dr. R. Khedri

Outline

Introduction


Basic Definitions

Atoms

Dense sets




(Slide 24 of 32)Boolean Algebras Representations of Booleanalgebras

It is quite straightforward to check that a field of setsobeys the axioms for Boolean algebra

No theory will be able to distinguish betweenisomorphic algebras, so we should not hope that onlyfields of sets form Boolean algebras

Surprisingly, the finitely many axioms for Booleanalgebra determine exactly when an algebra isisomorphic to a field of sets - when it is representable

Definition (Representable B.A.)

A Boolean algebra B is said to be representable if it isisomorphic to a field of sets. An isomorphism from aBoolean algebra B to a field of sets F is called arepresentation of B; its base is defined to be the base of F .



Applications

Dr. R. Khedri

Outline

Introduction


Basic Definitions

Atoms

Dense sets





Theorem (Stone theorem)

Every Boolean algebra is isomorphic to a field of sets.

Proof highlights:

Let B be any boolean algebra

The base of our field of sets is the set Uf(B) ofultrafilters of BThe embeddingh : B −→ F = 〈P(Uf(B)), ∅,Uf(B),∪, \〉h is defined by h(b) = {γ ∈ Uf(B) | b ∈ γ}Is it an embedding?

+ is preservedh is a bijection (use BPI)

Thus, B is isomorphic to the subalgebra of F withdomain ran (()h), and any subalgebra of a field of setsis also a field of sets



Applications

Dr. R. Khedri

Outline

Introduction


Basic Definitions

Atoms

Dense sets





Definition (Complete representation)

A complete representation h : B −→ F , where F is a fieldof sets, is a representation such that whenever S ⊆ B and+(S | S ⊆ B : S ) exists then

h(

+ (S | S ⊆ B : S ))

= ∪(s | s ∈ S : {h(s)} ).

A boolean algebra is said to be completelyrepresentable if it has a complete representation

A representation h of B is atomic if and only if it iscomplete



Applications

Dr. R. Khedri

Outline

Introduction


Basic Definitions

Atoms

Dense sets




(Slide 27 of 32)Boolean Algebras Complete Boolean Algebra

Theorem

A boolean algebra B has a complete representation if andonly if it is an atomic boolean algebra.

Definition (Complete B.A.)

A boolean algebra B is said to be complete if+(S | ⊆ B : S ) and ·(S | ⊆ B : S ) exist.

A countably infinite atomic boolean algebra cannot becomplete

Any finite boolean algebra is complete, as is a field ofsets of the form (P(X ), ∅,X ,∪, \)Every boolean algebra is a dense subalgebra of acomplete boolean algebra THE END



Applications

Dr. R. Khedri

Outline

Introduction


Basic Definitions

Atoms

Dense sets







Applications

Dr. R. Khedri

Outline

Introduction


Basic Definitions

Atoms

Dense sets




(Slide 32 of 32)

cas 738 relation algebra and kleene algebra and their ...€¦ · representations of boolean...

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