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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 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 "finite" in its normal set theoretic meaning does not coincide with the order-theoretic notion of a "finite element". == 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Compact element」の詳細全文を読む スポンサード リンク
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