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''.

Overview

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

:$x*\left(\bigvee_\right) = \bigvee_(x*y_i)$

and

:$\left(\bigvee_\right)*=\bigvee_(y_i*x)$

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 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:

:$\biggl(\bigvee_\biggr)^\circ =\bigvee_(x_i^\circ).$

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

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

:$f\left(\bigvee_\right) = \bigvee_ f(x_i).$

See also

* Relation algebra

References

* J. Paseka, J. Rosicky, Quantales, in: Bob Coecke, 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.
* M. Piazza, M. Castellan, ''Quantales and structural rules''. Journal of Logic and Computation, 6 (1996), 709–724.
* K. Rosenthal, ''Quantales and Their Applications'', Pitman Research Notes in Mathematics Series 234, Longman Scientific & Technical, 1990.

Order theory

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