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.
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:
- 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 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 involutive quantale is a quantale with an involution:
that preserves joins:
- C.J. Mulvey (2001), "Quantales", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 
- 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.