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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.

Overview

A quantale is a complete lattice ${\displaystyle Q}$ with an associative binary operation ${\displaystyle \ast \colon Q\times Q\to Q}$, called its multiplication, satisfying a distributive property such that

${\displaystyle x*\left(\bigvee _{i\in I}{y_{i}}\right)=\bigvee _{i\in I}(x*y_{i})}$

and

${\displaystyle \left(\bigvee _{i\in I}{y_{i}}\right)*{x}=\bigvee _{i\in I}(y_{i}*x)}$

for all ${\displaystyle x,y_{i}\in Q}$ and ${\displaystyle i\in I}$ (here ${\displaystyle I}$ is any index set). The quantale is unital if it has an identity element ${\displaystyle e}$ for its multiplication:

${\displaystyle x*e=x=e*x}$

for all ${\displaystyle x\in Q}$. In this case, the quantale is naturally a monoid with respect to its multiplication ${\displaystyle \ast }$.

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

${\displaystyle (xy)^{\circ }=y^{\circ }x^{\circ }}$

that preserves joins:

${\displaystyle {\biggl (}\bigvee _{i\in I}{x_{i}}{\biggr )}^{\circ }=\bigvee _{i\in I}(x_{i}^{\circ }).}$

A quantale homomorphism is a map ${\displaystyle f\colon Q_{1}\to Q_{2}}$ that preserves joins and multiplication for all ${\displaystyle x,y,x_{i}\in Q_{1}}$ and ${\displaystyle i\in I}$:

${\displaystyle f(xy)=f(x)f(y),}$

${\displaystyle f\left(\bigvee _{i\in I}{x_{i}}\right)=\bigvee _{i\in I}f(x_{i}).}$

See also

• Relation algebra

References

1. Paeska, Jan; Slesinger, Radek (2018). "A representation theorem for quantale valued sup-algebras". IEEE 48th International Symposium on Multiple-Valued Logic: 1 – via IEEE Xplore.

• C.J. Mulvey (2001) , "Quantale", Encyclopedia of Mathematics, EMS Press
• 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.

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