Formal Definition
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 nonempty 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 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 be that this is the only compact element, as the example of the real unit interval shows.
Read more about this topic: Compact Element
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