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 ...
algebraic structure
In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set ...
s that generalize locales (
point free topologies) as well as various multiplicative
lattices of
ideals from ring theory and functional analysis (
C*-algebras
In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continuou ...
,
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
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...
binary operation
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two.
More specifically, an internal binary op ...
∗ : ''Q'' × ''Q'' → ''Q'', called its multiplication, satisfying a distributive property such that
:
and
:
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 consis ...
). 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 ...
''e'' for its multiplication:
:
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.
Monoids a ...
with respect to its multiplication ∗.
A unital quantale may be defined equivalently as 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.
Monoids 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
In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined dually, that is, it is an elem ...
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
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 (co ...
, 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
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being addi ...
.
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 p ...
. 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 (co ...
is the same as an idempotent strictly two-sided quantale.
An involutive quantale is a quantale with an involution
:
that preserves joins:
:
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. Althoug ...
''f'' : ''Q
1'' → ''Q
2'' that preserves joins and multiplication for all ''x'', ''y'', ''x
i'' in ''Q
1'', and ''i'' in ''I'':
:
:
See also
*
Relation algebra
In mathematics and abstract algebra, a relation algebra is a residuated Boolean algebra expanded with an involution called converse, a unary operation. The motivating example of a relation algebra is the algebra 2''X''² of all binary relations ...
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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