词条 | Quantale |
释义 |
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. OverviewA quantale is a complete lattice Q with an associative binary operation ∗ : Q × Q → Q, called its multiplication, satisfying a distributive property such that- 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 Q1, i in I: See also
References
1 : Order theory |
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