# 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

and

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:

*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, 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 : *Q _{1}* →

*Q*that preserves joins and multiplication for all

_{2}*x*,

*y*,

*x*in

_{i}*Q*,

*i*in

*I*:

## References

- {{#invoke:citation/CS1|citation

|CitationClass=citation }} [1]

- J. Paseka, J. Rosicky, Quantales, in: B. Coecke, D. Moore, A. Wilce, (Eds.),
*Current Research in Operational Quantum Logic: Algebras, Categories and Languages*, Fund. Theories Phys., vol. 111, Kluwer Academic Publishers, 2000, pp. 245–262. - K. Rosenthal,
*Quantales and Their Applications*, Pitman Research Notes in Mathematics Series 234, Longman Scientific & Technical, 1990.