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In mathematics, particularly in order theory, a pseudocomplement is one generalization of the notion of complement. In a lattice ''L'' with bottom element 0, an element ''x'' ∈ ''L'' is said to have a ''pseudocomplement'' if there exists a greatest element ''x'' * ∈ ''L'', disjoint from ''x'', with the property that ''x'' ∧ ''x'' * = 0. More formally, ''x'' * = max. The lattice ''L'' itself is called a pseudocomplemented lattice if every element of ''L'' is pseudocomplemented. Every pseudocomplemented lattice is necessarily bounded, i.e. it has a 1 as well. Since the pseudocomplement is unique by definition (if it exists), a pseudocomplemented lattice can be endowed with a unary operation * mapping every element to its pseudocomplement; this structure is sometimes called a ''p''-algebra. However this latter term may have other meanings in other areas of mathematics. ==Properties== In a ''p''-algebra ''L'', for all ''x'', ''y'' ∈ ''L'':〔〔 * The map ''x'' ↦ ''x'' * is antitone. In particular, 0 * = 1 and 1 * = 0. * The map ''x'' ↦ ''x'' * * is a closure. * ''x'' * = ''x'' * * *. * (''x''∨''y'') * = ''x'' * ∧ ''y'' *. * (''x''∧''y'') * * = ''x'' * * ∧ ''y'' * *. The set ''S''(''L'') ≝ is called the skeleton of ''L''. ''S''(''L'') is a ∧-subsemilattice of ''L'' and together with ''x'' ∪ ''y'' = (''x''∨''y'') * * = (''x'' * ∧ ''y'' *) * forms a Boolean algebra (the complement in this algebra is *).〔〔 In general, ''S''(''L'') is not a sublattice of ''L''.〔 In a distributive ''p''-algebra, ''S''(''L'') is the set of complemented elements of L..〔 Every element ''x'' with the property ''x'' * = 0 (or equivalently, ''x'' * * = 1) is called dense. Every element of the form ''x'' ∨ ''x'' * is dense. ''D''(''L''), the set of all the dense elements in ''L'' is a filter of ''L''.〔〔 A distributive ''p''-algebra is Boolean if and only if ''D''(''L'') = .〔 Pseudocomplemented lattices form a variety.〔 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「Pseudocomplement」の詳細全文を読む スポンサード リンク
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