Compact element
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In the mathematical area of order theory, the compact or finite elements of a partially ordered set are those elements that cannot be subsumed by a supremum of any directed set that does not already contain members above the compact element. Note that there are other notions of compactness in mathematics and that the term finite with its common set theoretic semantics does not coincide with the order theoretic notion of finite elements either.
Compact elements are important in domain theory, where they are considered as a kind of primitive element: the information represented by compact elements cannot be obtained by any approximation that does not already contain this knowledge. Especially compact elements cannot be approximated by elements strictly below them. On the other hand, it may happen that all non-compact elements can be obtained as directed suprema of compact elements. This is a desirable situation, since the set of compacts is often smaller than the original poset -- the examples below illustrate this. Posets that can be recovered from their set of compact elements are called algebraic posets.
Formal definition
For some partially ordered set (P,≤) an element c of P is called compact (or finite) if it satisfies one of the following equivalent conditions:
- For every directed subset D of P, if D has a supremum sup D and c ≤ sup D then c ≤ d for some element d of D.
- For every ideal I of P, if I has a supremum sup I and c ≤ sup I than c is an element of I.
- The element c is way below itself, i.e. c << c
If the poset P additionally is a join-semilattice (i.e. if it has binary suprema) then these conditions are equivalent to the following statement:
- For every subset S of P, if S has a supremum sup S and c ≤ sup S, then c ≤ sup T for some finite subset T of S.
Using the definitions of the involved concepts these equivalences are easily verified. For the case of the join-semilattices note that any set can be turned into a directed set with the same supremum by closing under finite (non-empty) suprema.
When considering directed complete partial orders or complete lattices the additional requirements that the specified suprema exist can of course be dropped. Note also that a join-semilattice which is directed complete is almost a complete lattice (possibly lacking a least element) -- see completeness (order theory) for details.
If it exists, the least element of a poset is always compact. It may well be that this is the only compact element, as the example of the real unit interval [0,1] shows.
Examples
- The most basic example is obtained by considering the powerset of some set, ordered by subset inclusion. Within this complete lattice, the compact elements are exactly the finite sets. This justifies the name "finite element".
- The term "compact" is explained by considering the complete lattices of open sets of some topological space. Within this order, the compact elements are just the compact sets. Indeed, the condition for compactness in join-semilattices translates immediately to the corresponding definition.
Literature
Compact elements are standard. See the literature given for order theory and domain theory.