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.
When considering directed complete partial orders or complete lattices the additional requirements that the specified suprema exist can of course be dropped.
; or even a mere set without any operations), let Sub(A) be the set of all substructures of A, i.e., of all subsets of A which are closed under all operations of A (group addition, ring addition and multiplication, etc.).
Compact elements are important in computer science in the semantic approach called 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.
This is a desirable situation, since the set of compact elements is often smaller than the original poset—the examples above illustrate this.