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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 : and : 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: : that preserves joins: : A quantale homomorphism is a map f : ''Q1'' → ''Q2'' that preserves joins and multiplication for all ''x'', ''y'', ''xi'' in ''Q'', ''i'' in ''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
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