## Quantales

Abbreviation: Quant

### Definition

A \emph{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 $h(x \cdot y)=h(x) \cdot h(y)$

Example 1:

### Properties

Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described.

Classtype (value, see description) 1)

### Finite members

$\begin{array}{lr} f(1)= &1\\ f(2)= &2\\ f(3)= &12\\ f(4)= &129\\ f(5)= &1852\\ f(6)= &33391\\ \end{array}$

Model search done by Mace4 https://www.cs.unm.edu/~mccune/mace4/

### Subclasses

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

### Superclasses

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

### References

1) F. Lastname, \emph{Title}, Journal, \textbf{1}, 23–45 MRreview