relation algebra

relation algebra

It is a well-known fact that if

A

is a set, then

P

⁢

(

A

)

the

power set

of

A

, equipped with the

intersection

operation

∩

, the union operation

∪

, and the

complement

operation

′

turns

P

⁢

(

A

)

into a

Boolean algebra

. Indeed, a Boolean algebra can be viewed as an abstraction of the family of subsets of a set with the usual set-theoretic operations. The algebraic abstraction of the set of

binary relations

on a set is what is known as a

relation algebra

.

Before defining formally what a relation algebra is, let us recall the following facts about binary relations on some given set

A

:

•

binary relations are

closed under

∩

,

∪

and

′

, and in fact,

satisfy

all the axioms of a Boolean algebra. In short, the set

R

⁢

(

A

)

of binary relations on

A

is a Boolean algebra, where

A

×

A

is the top and

∅

×

∅

(

=

∅

)

is the bottom of

R

⁢

(

A

)

;

•

if

R

and

S

are binary relations on

A

, then so are

R

∘

S

,

relation composition

of

R

and

S

, and

R

-

1

, the

inverse

of

R

;

•

I

A

, defined by

{

(

a

,

a

)

∣

a

∈

A

}

, is a binary relation which is also the

identity

with respect to

∘

, also known as the

identity relation

on

A

;

•

some familiar identities:

(a)

R

∘

(

S

∘

T

)

=

(

R

∘

S

)

∘

T

(b)

(

R

-

1

)

-

1

=

R

(c)

(

R

∘

S

)

-

1

=

S

-

1

∘

R

-

1

(d)

(

R

∪

S

)

∘

T

=

(

R

∘

T

)

∪

(

S

∘

T

)

(e)

(

R

∪

S

)

-

1

=

R

-

1

∪

S

-

1

In

addition

, there is also a rule that is true for all binary relations on

A

:

(

R

∘

S

)

∩

T

=

∅

iff

(

R

-

1

∘

T

)

∩

S

=

∅

(1)

The verification of this rule is as follows: first note that

sufficiency

implies necessity, for if

(

R

-

1

∘

T

)

∩

S

=

∅

, then

(

R

∘

S

)

∩

T

=

(

(

R

-

1

)

-

1

∘

S

)

∩

T

=

∅

. To show sufficiency, suppose

(

a

,

b

)

∈

S

is also an element of

R

-

1

∘

T

. Then there is

c

∈

A

such that

(

a

,

c

)

∈

R

-

1

and

(

c

,

b

)

∈

T

. Therefore,

(

c

,

a

)

∈

R

and

(

c

,

b

)

=

(

c

,

a

)

∘

(

a

,

b

)

∈

R

∘

S

as well. This shows that

(

R

∘

S

)

∩

T

≠

∅

.

It can be shown that Rule (1) is

equivalent

to the following

inclusion

R

-

1

∘

(

R

∘

S

)

′

⊆

S

′

.

(2)

Definition

. A

relation algebra

is a Boolean algebra

B

with the usual

Boolean operators

∨

,

∧

,

′

, and additionally a

binary operator

;

, a unary

operator

-

, and a

constant

i

such that

1.

a

;

i

=

a

2.

a

;

(

b

;

c

)

=

(

a

;

b

)

;

c

3.

a

-

-

=

a

4.

(

a

;

b

)

-

=

b

-

;

a

-

5.

(

a

∨

b

)

;

c

=

(

a

;

c

)

∨

(

b

;

c

)

6.

(

a

∨

b

)

-

=

a

-

∨

b

-

7.

a

-

;

(

a

;

b

)

′

≤

b

′

where

≤

is the

induced

partial order

in the underlying Boolean algebra.

Clearly, the set of all binary relations

R

⁢

(

A

)

on a set

A

is a relation algebra, as we have just demonstrated. Specifically, in

R

⁢

(

A

)

,

;

is the composition operator

∘

,

-

is the inverse (or

converse

) operator

-

1

, and

i

is

I

A

.

A relation algebra is an

algebraic system

. As an algebraic system, we can define the usual algebraic notions, such as

subalgebras

of an

algebra

,

homomorphisms

between two algebras, etc… A relation algebra

B

that is a subalgebra of

R

⁢

(

A

)

, the set of all binary relations on a set

A

, is called a

set relation algebra

.

References

1

S. R. Givant,

The

Structure

of Relation Algebras Generated by Relativizations

, American Mathematical Society (1994).

Title

relation algebra

Canonical name

RelationAlgebra

Date of creation

2013-03-22 17:48:18

Last modified on

2013-03-22 17:48:18

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

11

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03G15

Classification

msc 06E25

Related topic

OperationsOnRelations

Defines

set relation algebra

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