In mathematics, quantales are certain

partially ordered In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary r ...

algebraic structures that generalize locales ( point free topologies) as well as various multiplicative lattices of

ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considere ...

s from ring theory and functional analysis ( C*-algebras,

von Neumann algebra In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type of C*-algebra. Von Neumann algebra ...

s). 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 In mathematics, an index set is a set whose members label (or index) members of another set. For instance, if the elements of a set may be ''indexed'' or ''labeled'' by means of the elements of a set , then is an index set. The indexing consists ...

). The quantale is unital if it has an

identity element In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...

''e'' for its multiplication: :

x*e = x = e*x

for all ''x'' in ''Q''. In this case, the quantale is naturally a

monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoid ...

with respect to its multiplication ∗. A unital quantale may be defined equivalently as a

in the category Sup of complete join semi-lattices. A unital quantale is an idempotent

semiring In abstract algebra, a semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse. The term rig is also used occasionally—this originated as a joke, suggesting that rigs ar ...

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 In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...

. A

frame A frame is often a structural system that supports other components of a physical construction and/or steel frame that limits the construction's extent. Frame and FRAME may also refer to: Physical objects In building construction *Framing (con ...

, 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 In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysis ...

together with its usual multiplication. An idempotent quantale is a quantale whose multiplication is

idempotent Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...

. A

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 In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...

is a

map A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes. Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interactive. Although ...

''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).

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 {{math-stub

Overview

See also

References