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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Quantale ： ウィキペディア英語版 Quantale In mathematics, quantales are certain partially ordered algebraic structures that generalize locales (point free topologies) as well as various multiplicative lattices of ideals from ring theory and functional analysis (C *-algebras, von Neumann algebras). Quantales are sometimes referred to as ''complete residuated semigroups''. A quantale is a complete lattice ''Q'' with an associative binary operation ∗ : ''Q'' × ''Q'' → ''Q'', called its multiplication, satisfying :$x$ *(\bigvee_)=\bigvee_(x *y_i) and :$(\backslash bigvee\_)$ *=\bigvee_(y_i *x) for all ''x'', ''yi'' in ''Q'', ''i'' in ''I'' (here ''I'' is any index set). The quantale is unital if it has an identity element ''e'' for its multiplication: : ''x'' ∗ ''e'' = ''x'' = ''e'' ∗ ''x'' for all ''x'' in ''Q''. In this case, the quantale is naturally a monoid with respect to its multiplication ∗. A unital quantale may be defined equivalently as a monoid in the category Sup of complete join semi-lattices. A unital quantale is an idempotent semiring, or dioid, under join and multiplication. A unital quantale in which the identity is the top element of the underlying lattice, is said to be strictly two-sided (or simply ''integral''). A commutative quantale is a quantale whose multiplication is commutative. A frame, with its multiplication given by the meet operation, is a typical example of a strictly two-sided commutative quantale. Another simple example is provided by the unit interval together with its usual multiplication. An idempotent quantale is a quantale whose multiplication is idempotent. A frame is the same as an idempotent strictly two-sided quantale. An involutive quantale is a quantale with an involution: :$(xy)^\backslash circ\; =\; y^\backslash circ\; x^\backslash circ$ that preserves joins: :$\backslash biggl(\backslash bigvee\_\backslash biggr)^\backslash circ\; =\backslash bigvee\_(x\_i^\backslash circ).$ A quantale homomorphism is a map f : ''Q1'' → ''Q2'' that preserves joins and multiplication for all ''x'', ''y'', ''xi'' in ''Q'', ''i'' in ''I'': :$f(xy)\; =\; f(x)f(y)$ :$f\backslash biggl(\backslash bigvee\_\backslash biggl)\; =\; \backslash bigvee\_\; f(x\_i)$ == See also == * relation algebra 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「Quantale」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.