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 Q {\displaystyle Q} with an associative binary operation ∗ : Q × Q → Q {\displaystyle \ast \colon Q\times Q\to Q} , called its multiplication, satisfying a distributive property such that
x ∗ ( ⋁ i ∈ I y i ) = ⋁ i ∈ I ( x ∗ y i ) {\displaystyle x*\left(\bigvee _{i\in I}{y_{i}}\right)=\bigvee _{i\in I}(x*y_{i})}
and
( ⋁ i ∈ I y i ) ∗ x = ⋁ i ∈ I ( y i ∗ x ) {\displaystyle \left(\bigvee _{i\in I}{y_{i}}\right)*{x}=\bigvee _{i\in I}(y_{i}*x)}
for all x , y i ∈ Q {\displaystyle x,y_{i}\in Q} and i ∈ I {\displaystyle i\in I} (here I {\displaystyle I} is any index set). The quantale is unital if it has an identity element e {\displaystyle e} for its multiplication:
x ∗ e = x = e ∗ x {\displaystyle x*e=x=e*x}
for all x ∈ Q {\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
( x y ) ∘ = y ∘ x ∘ {\displaystyle (xy)^{\circ }=y^{\circ }x^{\circ }}
that preserves joins:
( ⋁ i ∈ I x i ) ∘ = ⋁ i ∈ I ( x i ∘ ) . {\displaystyle {\biggl (}\bigvee _{i\in I}{x_{i}}{\biggr )}^{\circ }=\bigvee _{i\in I}(x_{i}^{\circ }).}
A quantale homomorphism is a map f : Q 1 → Q 2 {\displaystyle f\colon Q_{1}\to Q_{2}} that preserves joins and multiplication for all x , y , x i ∈ Q 1 {\displaystyle x,y,x_{i}\in Q_{1}} and i ∈ I {\displaystyle i\in I} :
f ( x y ) = f ( x ) f ( y ) , {\displaystyle f(xy)=f(x)f(y),}
f ( ⋁ i ∈ I x i ) = ⋁ i ∈ I f ( x i ) . {\displaystyle f\left(\bigvee _{i\in I}{x_{i}}\right)=\bigvee _{i\in I}f(x_{i}).}
See also Relation algebra
References
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