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.
Overview
A 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 ∗ xfor 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: