closure of a relation with respect to a property

closure of a relation with respect to a property

Introduction

Fix a set

A

. A

property

𝒫

of

n

-ary relations on a set

A

may be thought of as some subset of the set of all

n

-ary relations on

A

. Since an

n

-ary

relation

is just a subset of

A

n

,

𝒫

⊆

P

⁢

(

A

n

)

, the

powerset

of

A

n

. An

n

-ary relation is said to have property

𝒫

if

R

∈

𝒫

.

For example, the

transitive property

is a property of

binary relations

on

A

; it consists of all

transitive

binary relations on

A

.

Reflexive

and

symmetric

properties are sets of reflexive and symmetric binary relations on

A

correspondingly.

Let

R

be an

n

-ary relation on

A

. By the

closure

of an

n

-ary relation

R

with respect to property

𝒫

, or the

𝒫

-closure of

R

for short, we mean the smallest relation

S

∈

𝒫

such that

R

⊆

S

. In other words, if

T

∈

𝒫

and

R

⊆

T

, then

S

⊆

T

. We write

Cl

𝒫

⁡

(

R

)

for the

𝒫

-closure of

R

.

Given an

n

-ary relation

R

on

A

, and a property

𝒫

on

n

-ary relations on

A

, does

Cl

𝒫

⁡

(

R

)

always exist? The answer is no. For example, let

𝒫

be the anti-symmetric property of binary relations on

A

, and

R

=

A

2

. For another example, take

𝒫

to be the

irreflexive

property, and

R

=

Δ

, the

diagonal relation

on

A

.

However, if

A

n

∈

𝒫

and

𝒫

is

closed under

arbitrary

intersections

, then

𝒫

is a

complete lattice

according to this fact (

http://planetmath.org/CriteriaForAPosetToBeACompleteLattice

), and, as a result,

Cl

𝒫

⁡

(

R

)

exists for any

R

⊆

A

n

.

Reflexive, Symmetric, and Transitive Closures

From now on, we concentrate on binary relations on a set

A

. In particular, we fix a binary relation

R

on

A

, and let

𝒳

the reflexive property,

𝒮

the symmetric property, and

𝒯

be the transitive property on the binary relations on

A

.

Proposition 1

.

Arbitrary intersections are closed in

X

,

S

, and

T

. Furthermore, if

R

is any binary relation on

A

, then

•

R

=

:=

Cl

𝒳

⁡

(

R

)

=

R

∪

Δ

, where

Δ

is the diagonal relation on

A

,

•

R

↔

:=

Cl

𝒮

⁡

(

R

)

=

R

∪

R

-

1

, where

R

-

1

is the

converse

of

R

, and

•

R

+

:=

Cl

𝒯

⁡

(

R

)

is given by

⋃

n

∈

ℕ

R

n

=

R

∪

(

R

∘

R

)

∪

⋯

∪

(

R

∘

⋯

∘

R

)

⏟

n

⁢

-fold power

∪

⋯

,

where

∘

is the

relational composition

operator

.

•

R

*

:=

R

=

+

=

R

+

=

.

R

=

,

R

↔

,

R

+

, and

R

*

are called the

reflexive closure

, the

symmetric closure

, the

transitive closure

, and the

reflexive transitive closure

of

R

respectively. The last item in the

proposition

permits us to call

R

*

the

transitive reflexive closure

of

R

as well (there is no

difference

to the order of taking closures). This is true because

Δ

is transitive.

Remark

. In general, however, the order of taking closures of a relation is important. For example, let

A

=

{

a

,

b

}

, and

R

=

{

(

a

,

b

)

}

. Then

R

↔

+

=

A

2

≠

{

(

a

,

b

)

,

(

b

,

a

)

}

=

R

+

↔

.

Title

closure of a relation with respect to a property

Canonical name

ClosureOfARelationWithRespectToAProperty

Date of creation

2013-03-22 17:36:26

Last modified on

2013-03-22 17:36:26

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

15

Author

CWoo (3771)

Entry type

Definition

Classification

msc 08A02

Related topic

Property2

Defines

reflexive closure

Defines

symmetric closure

Defines

transitive closure

Defines

reflexive transitive closure

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