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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.

A quantale is a complete lattice Q with an associative binary operation ∗ : Q × Q → Q, called its multiplication, satisfying

$x*(\underset{i\in I}{\bigvee }{y}_i)=\underset{i\in I}{\bigvee }(x*y_i)$

and

$(\underset{i\in I}{\bigvee }{y}_i)*x=\underset{i\in I}{\bigvee }(y_i*x)$

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:

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:

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

that preserves joins:

$(\underset{i\in I}{\bigvee }{x}_i{)}^{\circ }=\underset{i\in I}{\bigvee }(x_i^{\circ }).$

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

$f(xy)=f(x)f(y)$

$f\left(\underset{i\in I}{\bigvee }{x}_i\right)=\underset{i\in I}{\bigvee }f(x_i)$

References

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• 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.