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 <math>Q</math> with an associative binary operation <math>\ast\colon Q \times Q \to Q</math>, called its multiplication, satisfying a distributive property such that
:<math>x*\left(\bigvee_{i\in I}{y_i}\right) = \bigvee_{i\in I}(x*y_i)</math>
and
:<math>\left(\bigvee_{i\in I}{y_i}\right)*{x}=\bigvee_{i\in I}(y_i*x)</math>
for all <math>x, y_i \in Q</math> and <math>i \in I</math> (here <math>I</math> is any index set). The quantale is unital if it has an identity element <math>e</math> for its multiplication:
:<math>x*e = x = e*x</math>
for all <math>x \in Q</math>. In this case, the quantale is naturally a monoid with respect to its multiplication <math>\ast</math>.
A unital quantale may be defined equivalently as a monoid in the category Sup of complete join-semilattices.
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
:<math>(xy)^\circ = y^\circ x^\circ</math>
that preserves joins:
:<math>\biggl(\bigvee_{i\in I}{x_i}\biggr)^\circ =\bigvee_{i\in I}(x_i^\circ).</math>
A quantale homomorphism is a map <math>f\colon Q_1 \to Q_2</math> that preserves joins and multiplication for all <math>x, y, x_i \in Q_1</math> and <math>i \in I</math>:
:<math>f(xy) = f(x) f(y),</math>
:<math>f\left(\bigvee_{i \in I}{x_i}\right) = \bigvee_{i \in I} f(x_i).</math>
See also
- Relation algebra
References
- [http://encyclopediaofmath.org/index.php?title=Quantale&oldid=42430]
- 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.
- 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.