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Concept
Compact element
Summary
In the mathematical area of order theory, the compact elements or finite elements of a partially ordered set are those elements that cannot be subsumed by a supremum of any non-empty directed set that does not already contain members above the compact element. This notion of compactness simultaneously generalizes the notions of finite sets in set theory, compact sets in topology, and finitely generated modules in algebra. (There are other notions of compactness in mathematics.)
In a partially ordered set (P,≤) an element c 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 then c is an element of I.
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.
In particular, if c = sup S, then c is the supremum of a finite subset of S.
These equivalences are easily verified from the definitions of the concepts involved. For the case of a join-semilattice, 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. A join-semilattice that is directed complete is almost a complete lattice (possibly lacking a least element)—see completeness (order theory) for details.
The most basic example is obtained by considering the power set of some set A, ordered by subset inclusion. Within this complete lattice, the compact elements are exactly the finite subsets of A. This justifies the name "finite element".
The term "compact" is inspired by the definition of (topologically) compact subsets of a topological space T.
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