Quantale
Encyclopedia
In mathematics
, quantales are certain partially ordered
algebraic structure
s that generalize locales (point free topologies
) as well as various multiplicative lattices
of ideal
s from ring theory and functional analysis (C*-algebras, von Neumann algebra
s). Quantales are sometimes referred to as complete residuated semigroups.
A quantale is a complete lattice
Q with an associative binary operation
∗ : Q × Q → Q, called its multiplication, satisfying
and
for all x, yi in Q, i in I (here I is any index set
).
The quantale is unital if it is has an identity element
e for its multiplication:
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:
that preserves joins:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, quantales are certain partially ordered
Partially ordered set
In mathematics, especially order theory, a partially ordered set 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 relation that indicates that, for certain pairs of elements in the...
algebraic structure
Algebraic structure
In abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sorts, closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties...
s that generalize locales (point free topologies
Pointless topology
In mathematics, pointless topology is an approach to topology that avoids mentioning points. The name 'pointless topology' is due to John von Neumann...
) as well as various multiplicative lattices
Lattice (order)
In mathematics, a lattice is a partially ordered set in which any two elements have a unique supremum and an infimum . Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities...
of ideal
Ideal
-In philosophy:* Ideal , values that one actively pursues as goals* Platonic ideal, a philosophical idea of trueness of form, associated with Plato-In mathematics:* Ideal , special subsets of a ring considered in abstract algebra...
s from ring theory and functional analysis (C*-algebras, von Neumann algebra
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. They were originally introduced by John von Neumann, motivated by his study of single operators, group...
s). Quantales are sometimes referred to as complete residuated semigroups.
A quantale is a complete lattice
Complete lattice
In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum and an infimum . Complete lattices appear in many applications in mathematics and computer science...
Q with an associative binary operation
Binary operation
In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Examples include the familiar arithmetic operations of addition, subtraction, multiplication and division....
∗ : Q × Q → Q, called its multiplication, satisfying
and
for all x, yi in Q, i in I (here I is any index set
Index set
In mathematics, the elements of a set A may be indexed or labeled by means of a set J that is on that account called an index set...
).
The quantale is unital if it is has an identity element
Identity element
In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...
e for its multiplication:
- x ∗ e = x = e ∗ x
for all x in Q. In this case, the quantale is naturally a monoid
Monoid
In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are naturally semigroups with identity. Monoids occur in several branches of mathematics; for...
with respect to its multiplication ∗.
A unital quantale may be defined equivalently as a monoid
Monoid (category theory)
In category theory, a monoid in a monoidal category is an object M together with two morphisms* \mu : M\otimes M\to M called multiplication,* and \eta : I\to M called unit,...
in the category Sup of complete join semi-lattices.
A unital quantale is an idempotent semiring
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...
, or dioid, under join and multiplication.
A unital quantale in which the identity is the top element
Greatest element
In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set is an element of S which is greater than or equal to any other element of S. The term least element is defined dually...
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
Complete Heyting algebra
In mathematics, especially in order theory, a complete Heyting algebra is a Heyting algebra which is complete as a lattice. Complete Heyting algebras are the objects of three different categories; the category CHey, the category Loc of locales, and its opposite, the category Frm of frames...
, with its multiplication given by the meet
Meet (mathematics)
In mathematics, join and meet are dual binary operations on the elements of a partially ordered set. A join on a set is defined as the supremum with respect to a partial order on the set, provided a supremum exists...
operation, is a typical example of a strictly two-sided commutative quantale. Another simple example is provided by the unit interval
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...
together with its usual multiplication
Multiplication
Multiplication is the mathematical operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic ....
.
An idempotent quantale is a quantale whose multiplication is idempotent. A frame
Complete Heyting algebra
In mathematics, especially in order theory, a complete Heyting algebra is a Heyting algebra which is complete as a lattice. Complete Heyting algebras are the objects of three different categories; the category CHey, the category Loc of locales, and its opposite, the category Frm of frames...
is the same as an idempotent strictly two-sided quantale.
An involutive quantale is a quantale with an involution:
that preserves joins: