Abbreviation: Quant

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 $x\cdot 0=0=0\cdot x$.

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:

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.

$\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/

[[...]] subvariety

[[...]] expansion

[[...]] supervariety

[[...]] subreduct