compact element


Let X be a set and 𝒯 be a topologyMathworldPlanetmath 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 latticeMathworldPlanetmath. 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 collectionMathworldPlanetmath of closed subsets of X, and partial orderMathworldPlanetmath 𝒟 by inclusion, then 𝒟 becomes a lattice with meet and join defined by set theoretic intersectionMathworldPlanetmathPlanetmath 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. 1.

    Let C be a set and 2C the subset lattice (power setMathworldPlanetmath) of C. The compact elements of 2C are the finite subsets of C.

  2. 2.

    Let V be a vector space and L⁢(V) be the subspaceMathworldPlanetmathPlanetmath lattice of V. Then the compact elements of L⁢(V) are exactly the finite dimensional subspaces of V.

  3. 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. 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 numbersMathworldPlanetmath (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, ContinuousPlanetmathPlanetmath 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