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).
[1] Quantales are sometimes referred to as complete residuated semigroups.
, called its multiplication, satisfying a distributive property such that and for all
The quantale is unital if it has an identity element
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-semilattices.
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 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.
A frame is the same as an idempotent strictly two-sided quantale.