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

${\displaystyle x*\left(\bigvee _{i\in I}{y_{i))\right)=\bigvee _{i\in I}(x*y_{i})}$

and

${\displaystyle \left(\bigvee _{i\in I}{y_{i))\right)*{x}=\bigvee _{i\in I}(y_{i}*x)}$

for all x, y_i 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:

${\displaystyle x*e=x=e*x}$

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 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

${\displaystyle (xy)^{\circ }=y^{\circ }x^{\circ ))$

that preserves joins:

${\displaystyle {\biggl (}\bigvee _{i\in I}{x_{i)){\biggr )}^{\circ }=\bigvee _{i\in I}(x_{i}^{\circ }).}$

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

${\displaystyle f(xy)=f(x)f(y),}$

${\displaystyle f\left(\bigvee _{i\in I}{x_{i))\right)=\bigvee _{i\in I}f(x_{i}).}$