Abbreviation: OSlat

An \emph{ordered semilattice} is a ordered semigroup $\mathbf{A}=\langle A,\cdot,\le\rangle$ that is

\emph{commutative}: $x\cdot y = y\cdot x$ and

\emph{idempotent}: $x\cdot x = x$

Let $\mathbf{A}$ and $\mathbf{B}$ be ordered semigroups. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a orderpreserving homomorphism: $h(x \cdot y)=h(x) \cdot h(y)$, $x\le y\Longrightarrow h(x)\le h(y)$.

Example 1:

$\begin{array}{rr} f(1)=&1 f(2)=&2 f(3)=&5 f(4)=&14 f(5)=&42 f(6)=&132 f(7)=& f(8)=& \end{array}$

This sequence is the Catalan numbers http://oeis.org/A000108

Commutative ordered semigroups