closure of a subset under relations

closure of a subset under relations

Let

A

be a set and

R

be an

n

-ary relation on

A

,

n

≥

1

. A subset

B

of

A

is said to be

closed under

R

, or

R

-closed

, if, whenever

b

1

,

…

,

b

n

-

1

∈

B

, and

(

b

1

,

…

,

b

n

-

1

,

b

n

)

∈

R

, then

b

n

∈

B

.

Note that if

R

is unary, then

B

is

R

-closed iff

R

⊆

B

.

More generally, let

A

be a set and

ℛ

a set of (finitary) relations on

A

. A subset

B

is said to be

R

-closed

if

B

is

R

-closed for each

R

∈

ℛ

.

Given

B

⊆

A

with a set of

relations

ℛ

on

A

, we say that

C

⊆

A

is an

R

-

closure

of

B

if

1.

B

⊆

C

,

2.

C

is

ℛ

-closed, and

3.

if

D

⊆

A

satisfies

both 1 and 2, then

C

⊆

D

.

By condition 3,

C

, if exists, must be unique. Let us call it the

ℛ

-closure of

B

, and denote it by

Cl

ℛ

⁡

(

B

)

. If

ℛ

=

{

R

}

, then we call it the

R

-closure of

B

, and denote it by

Cl

R

⁡

(

B

)

correspondingly.

Here are some examples.

1.

Let

A

=

ℤ

,

B

=

{

5

}

, and

R

be the relation that

m

⁢

R

⁢

n

whenever

m

divides

n

. Clearly

B

is not closed under

R

(for example,

(

5

,

10

)

∈

R

but

10

∉

B

). Then

Cl

R

⁡

(

B

)

=

5

⁢

ℤ

. If

R

is instead the relation

≤

, then

Cl

≤

⁡

(

B

)

=

{

n

∈

A

∣

n

≥

5

}

.

2.

This is an example where

R

is in fact a

function

(

operation

). Suppose

A

=

ℤ

and

R

is the

binary operation

subtraction

-

. Suppose

B

=

{

3

,

5

}

. Then

Cl

-

⁡

(

B

)

=

A

. To see this, set

C

=

Cl

-

⁡

(

B

)

. Note that

2

=

5

-

3

∈

C

so

1

=

3

-

2

as well as

-

1

=

2

-

3

∈

C

. This means that if

n

∈

C

, both

n

-

1

and

n

+

1

∈

C

. By

induction

,

C

=

ℤ

. In general, if

B

=

{

p

,

q

}

, where

p

,

q

are

coprime

, then

Cl

-

⁡

(

B

)

=

ℤ

. This is essentially the result of the

Chinese Remainder Theorem

.

3.

If

R

is unary, then the

R

-closure of

B

⊆

A

is just

B

∪

R

. When every

R

∈

ℛ

is unary, then the

ℛ

-closure of

B

in

A

is

(

⋃

ℛ

)

∪

B

.

Proposition 1

.

Cl

ℛ

⁡

(

B

)

exists for every

B

⊆

A

.

Proof.

Let

S

ℛ

⁢

(

B

)

be the set of subsets of

A

satisfying the defining conditions 1 and 2 of

ℛ

-closures above, partially ordered by

⊆

. If

𝒞

⊆

S

ℛ

⁢

(

B

)

, then

⋂

𝒞

∈

S

ℛ

⁢

(

B

)

. To see this, we break the statement down into cases:

•

In the case when

𝒞

=

∅

, we have

⋂

𝒞

=

A

∈

S

ℛ

⁢

(

B

)

.

•

When

C

:=

⋂

𝒞

≠

∅

, pick any

n

-ary relation

R

∈

ℛ

.

(a)

If

n

=

1

, then, since aach

D

∈

𝒞

is

R

-closed,

R

⊆

D

. Therefore,

R

⊆

⋂

𝒞

=

⋂

{

D

∣

D

∈

𝒞

}

=

C

. So

C

is

R

-closed.

(b)

If

n

>

1

, pick

elements

c

1

,

…

,

c

n

-

1

∈

C

such that

(

c

1

,

…

,

c

n

)

∈

R

. As each

c

i

∈

D

for

i

=

1

,

…

,

n

-

1

, and

D

is

R

-closed,

c

n

∈

D

. Since

c

n

∈

D

for every

D

∈

𝒞

,

c

n

∈

C

as well. This shows that

C

is

R

-closed.

In both cases,

B

⊆

C

since

B

⊆

D

for every

D

∈

𝒞

. Therefore,

C

∈

S

ℛ

⁢

(

B

)

.

Hence,

S

ℛ

⁢

(

B

)

is a

complete lattice

by virtue of this fact (

http://planetmath.org/CriteriaForAPosetToBeACompleteLattice

), which means that

S

ℛ

⁢

(

B

)

has a

minimal element

, which is none other than the

ℛ

-closure

Cl

ℛ

⁡

(

B

)

of

B

.
∎

Remark

. It is not hard to see

Cl

ℛ

has the following properties:

1.

B

⊆

Cl

ℛ

⁡

(

B

)

,

2.

Cl

ℛ

⁡

(

Cl

ℛ

⁡

(

B

)

)

=

Cl

ℛ

⁡

(

B

)

, and

3.

if

B

⊆

C

, then

Cl

ℛ

⁡

(

B

)

⊆

Cl

ℛ

⁡

(

C

)

.

Next, assume that

𝒮

is another set of finitary relations on

A

. Then

1.

if

ℛ

⊆

𝒮

, then

Cl

𝒮

⁡

(

B

)

⊆

Cl

ℛ

⁡

(

B

)

,

2.

Cl

𝒮

⁡

(

Cl

ℛ

⁡

(

B

)

)

⊆

Cl

ℛ

∩

𝒮

⁡

(

B

)

, and

3.

Cl

𝒮

⁡

(

Cl

ℛ

⁡

(

B

)

)

=

Cl

ℛ

∩

𝒮

⁡

(

B

)

if

ℛ

⊆

𝒮

or

𝒮

⊆

ℛ

.

Title

closure of a subset under relations

Canonical name

ClosureOfASubsetUnderRelations

Date of creation

2013-03-22 16:21:26

Last modified on

2013-03-22 16:21:26

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

36

Author

CWoo (3771)

Entry type

Definition

Classification

msc 08A02

Defines

closed under

Defines

closure property

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