Quantale

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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 × QQ, called its multiplication, satisfying

x*(\bigvee_{i\in I}{y_i})=\bigvee_{i\in I}(x*y_i)

and

(\bigvee_{i\in I}{y_i})*{x}=\bigvee_{i\in I}(y_i*x)

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:

xe = x = ex

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:

\biggl(\bigvee_{i\in I}{x_i}\biggr)^\circ =\bigvee_{i\in I}(x_i^\circ).

A quantale homomorphism is a map f : Q1Q2 that preserves joins and multiplication for all x, y, xi in Q, i in I:

f(xy) = f(x)f(y)
f\biggl(\bigvee_{i \in I}{x_i}\biggl) = \bigvee_{i \in I} f(x_i)

References[edit]

  • C.J. Mulvey (2001), "Quantales", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4  [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.