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In mathematics, especially in order theory, a complete Heyting algebra is a Heyting algebra that is complete as a lattice. Complete Heyting algebras are the objects of three different categories; the category CHey, the category Loc of locales, and its opposite, the category Frm of frames. Although these three categories contain the same objects, they differ in their morphisms, and thus get distinct names. Only the morphisms of CHey are homomorphisms of complete Heyting algebras. Locales and frames form the foundation of pointless topology, which, instead of building on point-set topology, recasts the ideas of general topology in categorical terms, as statements on frames and locales. == Definition == Consider a partially ordered set (''P'', ≤) that is a complete lattice. Then ''P'' is a ''complete Heyting algebra'' if any of the following equivalent conditions hold: * ''P'' is a Heyting algebra, i.e. the operation has a right adjoint (also called the lower adjoint of a (monotone) Galois connection), for each element ''x'' of ''P''. * For all elements ''x'' of ''P'' and all subsets ''S'' of ''P'', the following infinite distributivity law holds: : * ''P'' is a distributive lattice, i.e., for all ''x'', ''y'' and ''z'' in ''P'', we have : : and the meet operations are Scott continuous for all ''x'' in ''P'' (i.e., preserve the suprema of directed sets) . 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「Complete Heyting algebra」の詳細全文を読む スポンサード リンク
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