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 a distributive property such that


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:

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 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 : Q1 Q2 that preserves joins and multiplication for all x, y, xi in Q1, and i in I:

See also


  • C.J. Mulvey (2001) [1994], "Quantales", in Hazewinkel, Michiel (ed.), Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, 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.
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.