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In the mathematical discipline of order theory, a complemented lattice is a bounded lattice (with least element 0 and greatest element 1), in which every element ''a'' has a complement, i.e. an element ''b'' satisfying ''a'' ∨ ''b'' = 1 and ''a'' ∧ ''b'' = 0. Complements need not be unique. A relatively complemented lattice is a lattice such that every interval (), viewed as a bounded lattice in its own right, is a complemented lattice. An orthocomplementation on a complemented lattice is an involution which is order-reversing and maps each element to a complement. An orthocomplemented lattice satisfying a weak form of the modular law is called an orthomodular lattice. In distributive lattices, complements are unique. Every complemented distributive lattice has a unique orthocomplementation and is in fact a Boolean algebra. ==Definition and basic properties== A complemented lattice is a bounded lattice (with least element 0 and greatest element 1), in which every element ''a'' has a complement, i.e. an element ''b'' such that ::''a'' ∨ ''b'' = 1 and ''a'' ∧ ''b'' = 0. In general an element may have more than one complement. However, in a (bounded) distributive lattice every element will have at most one complement.〔Grätzer (1971), Lemma I.6.1, p. 47. Rutherford (1965), Theorem 9.3 p. 25.〕 A lattice in which every element has exactly one complement is called a uniquely complemented lattice〔.〕 A lattice with the property that every interval (viewed as a sublattice) is complemented is called a relatively complemented lattice. In other words, a relatively complemented lattice is characterized by the property that for every element ''a'' in an interval (''d'' ) there is an element ''b'' such that ::''a'' ∨ ''b'' = ''d'' and ''a'' ∧ ''b'' = ''c''. Such an element ''b'' is called a complement of ''a'' relative to the interval. A distributive lattice is complemented if and only if it is bounded and relatively complemented.〔Grätzer (1971), Lemma I.6.2, p. 48. This result holds more generally for modular lattices, see Exercise 4, p. 50.〕〔Birkhoff (1961), Corollary IX.1, p. 134〕 The lattice of subspaces of a vector space provide an example of a complemented lattice that is not, in general, distributive. 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「Complemented lattice」の詳細全文を読む スポンサード リンク
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