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