Quantale

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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.

Overview

A quantale is a complete lattice Q with an associative binary operation ∗ : Q × QQ, called its multiplication, satisfying a distributive property such that

[math]\displaystyle{ x*\left(\bigvee_{i\in I}{y_i}\right) = \bigvee_{i\in I}(x*y_i) }[/math]

and

[math]\displaystyle{ \left(\bigvee_{i\in I}{y_i}\right)*{x}=\bigvee_{i\in I}(y_i*x) }[/math]

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:

[math]\displaystyle{ x*e = x = e*x }[/math]

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

[math]\displaystyle{ (xy)^\circ = y^\circ x^\circ }[/math]

that preserves joins:

[math]\displaystyle{ \biggl(\bigvee_{i\in I}{x_i}\biggr)^\circ =\bigvee_{i\in I}(x_i^\circ). }[/math]

A quantale homomorphism is a map f : Q1Q2 that preserves joins and multiplication for all x, y, xi in Q1, and i in I:

[math]\displaystyle{ f(xy) = f(x)f(y), }[/math]
[math]\displaystyle{ f\left(\bigvee_{i \in I}{x_i}\right) = \bigvee_{i \in I} f(x_i). }[/math]

See also

References