What is "quantale"

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

x * (⋁y) = ⋁(x * y)

and

(⋁y) * x = ⋁(y * x)

for all x, y 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:

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:

(xy) = yx
that preserves joins:

$$\biggl(\bigvee_{i\in I}{x_i}\biggr)^\circ =\bigvee_{i\in I}(x_i^\circ).$$

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

f(xy) = f(x)f(y)

$$f\biggl(\bigvee_{i \in I}{x_i}\biggl) = \bigvee_{i \in I} f(x_i)$$