Abbreviation: OSlat

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

commutative: $x\cdot y = y\cdot x$ and

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