Compact element

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 non-empty directed set that does not already contain members above the compact element.

Note that there are other notions of compactness in mathematics; also, the term "" in its normal set theoretic meaning does not coincide with the order-theoretic notion of a "finite element".

In a partially ordered set (P,≤) an element c is called compact (or finite) if it satisfies one of the following equivalent conditions:

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:

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 [0,1] shows.

A poset in which every element is the supremum of the compact elements below it is called an algebraic poset. Such posets which are dcpos are much used in domain theory.

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