compact element

compact element

Let

X

be a set and

𝒯

be a

topology

on

X

, a well-known

concept

is that of a

compact set

: a set

A

is

compact

if every

open cover

of

A

has a finite subcover. Another way of putting this, symbolically, is that if

A

⊆

⋃

𝒮

,

where

𝒮

⊂

𝒯

, then there is a finite subset

ℱ

of

𝒮

, such that

A

⊆

⋃

ℱ

.

A more general concept, derived from above, is that of a

compact element

in a

lattice

. Let

L

be a lattice and

a

∈

L

. Then

a

is said to be

compact

if

whenever a subset

S

of

L

such that

⋁

S

exists and

a

≤

⋁

S

, then there is a finite subset

F

⊂

S

such that

a

≤

⋁

F

.

If we let

𝒟

to be the

collection

of

closed subsets

of

X

, and

partial order

𝒟

by

inclusion

, then

𝒟

becomes a lattice with meet and join defined by set theoretic

intersection

and union. It is easy to see that an

element

A

∈

𝒟

is a compact element iff

D

is a compact closed subset in

X

.

Here are some other common examples:

1.

Let

C

be a set and

2

C

the subset lattice (

power set

) of

C

. The compact elements of

2

C

are the finite subsets of

C

.

2.

Let

V

be a

vector space

and

L

⁢

(

V

)

be the

subspace

lattice of

V

. Then the compact elements of

L

⁢

(

V

)

are exactly the finite dimensional subspaces of

V

.

3.

Let

G

be a group and

L

⁢

(

G

)

the

subgroup lattice

of

G

. Then the compact elements are the

finitely generated subgroups

of

G

.

4.

Note in all of the above examples, atoms are compact. However, this is not true in general. Let’s

construct

one such example. Adjoin the symbol

∞

to the lattice

ℕ

of

natural numbers

(with

linear order

), so that

n

<

∞

for all

n

∈

ℕ

. So

∞

is the top element of

ℕ

∪

{

∞

}

(and

1

is the bottom element!). Next, adjoin a symbol

a

to

ℕ

∪

{

∞

}

, and define the meet and join properties with

a

by

–

a

∨

n

=

∞

,

a

∧

n

=

1

for all

n

∈

ℕ

, and

–

a

∨

∞

=

∞

,

a

∧

∞

=

a

.

The resulting set

L

=

ℕ

∪

{

∞

,

a

}

is a lattice where

a

is a non-compact atom.

Remarks

.

•

As we have seen from the examples above,

compactness

is closely associated with the concept of finiteness, a compact element is sometimes called a

finite element

.

•

Any finite join of compact elements is compact.

•

An element

a

in a lattice

L

is compact iff for any directed (

http://planetmath.org/DirectedSet

) subset

D

of

L

such that

⋁

D

exists and

a

≤

⋁

D

, then there is an element

d

∈

D

such that

a

≤

d

.

•

As the last example indicates, not all atoms are compact. However, in an

algebraic lattice

, atoms are compact. The first three examples are all

instances

of algebraic lattices.

•

A compact element may be defined in an arbitrary poset

P

:

a

∈

P

is compact iff

a

is way below itself:

a

≪

a

.

References

1

G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. W. Mislove, D. S. Scott,

Continuous

Lattices and Domains

, Cambridge University Press, Cambridge (2003).

Title

compact element

Canonical name

CompactElement

Date of creation

2013-03-22 15:52:50

Last modified on

2013-03-22 15:52:50

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

17

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06B23

Synonym

finite element

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