Quantales

Abbreviation: Quant

Definition

A quantale is a structure $\mathbf{A}=\langle A, \bigvee, \cdot, 0\rangle$ of type $\langle\infty, 2, 0\rangle$ such that

$\langle A, \bigvee, 0\rangle$ is a complete semilattice with $0=\bigvee\emptyset$,

$\langle A, \cdot\rangle$ is a semigroup, and

$\cdot$ distributes over $\bigvee$: $(\bigvee X)\cdot y=\bigvee_{x\in X}(x\cdot y)$ and $y\cdot(\bigvee X)=\bigvee_{x\in X}(y\cdot x)$

Remark: In particular, $\cdot$ distributes over the empty join, so $x\cdot 0=0=0\cdot x$.

Morphisms

Let $\mathbf{A}$ and $\mathbf{B}$ be quantales. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: $h(\bigvee X)=\bigvee h[X]$ for all $X\subseteq A$ (hence $h(0)=0$) and $h(x \cdot y)=h(x) \cdot h(y)$

Examples

Example 1:

Basic results

Properties

Finite members

$\begin{array}{lr} f(1)= &1\\ f(2)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\ \end{array}$ $\begin{array}{lr} f(6)= &\\ f(7)= &\\ f(8)= &\\ f(9)= &\\ f(10)= &\\ \end{array}$

Subclasses

[[...]] subvariety
[[...]] expansion

Superclasses

[[...]] supervariety
[[...]] subreduct

References


1) F. Lastname, Title, Journal, 1, 23–45 MRreview