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In mathematics, a distributive lattice is a lattice in which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set union and intersection. Indeed, these lattices of sets describe the scenery completely: every distributive lattice is – up to isomorphism – given as such a lattice of sets. ==Definition== As in the case of arbitrary lattices, one can choose to consider a distributive lattice ''L'' either as a structure of order theory or of universal algebra. Both views and their mutual correspondence are discussed in the article on lattices. In the present situation, the algebraic description appears to be more convenient: A lattice (''L'',∨,∧) is distributive if the following additional identity holds for all ''x'', ''y'', and ''z'' in ''L'': : ''x'' ∧ (''y'' ∨ ''z'') = (''x'' ∧ ''y'') ∨ (''x'' ∧ ''z''). Viewing lattices as partially ordered sets, this says that the meet operation preserves non-empty finite joins. It is a basic fact of lattice theory that the above condition is equivalent to its dual:〔; here: §5-6, p.8-12〕 : ''x'' ∨ (''y'' ∧ ''z'') = (''x'' ∨ ''y'') ∧ (''x'' ∨ ''z'') for all ''x'', ''y'', and ''z'' in ''L''.〔For individual elements ''x'', ''y'', ''z'', e.g. the first equation may be violated, but the second may hold; see the N5 picture for an example.〕 In every lattice, defining ''p''≤''q'' as usual to mean ''p''∧''q''=''p'', the inequation ''x'' ∧ (''y'' ∨ ''z'') ≥ (''x'' ∧ ''y'') ∨ (''x'' ∧ ''z'') holds as well as its dual inequation ''x'' ∨ (''y'' ∧ ''z'') ≤ (''x'' ∨ ''y'') ∧ (''x'' ∨ ''z''). A lattice is distributive if one of the converse inequations holds, too. More information on the relationship of this condition to other distributivity conditions of order theory can be found in the article on distributivity (order theory). 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「Distributive lattice」の詳細全文を読む スポンサード リンク
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