Abbreviation: IOMon
An \emph{integral ordered monoid} is a ordered monoid $\mathbf{A}=\langle A,\cdot,1,\le\rangle$ that is
\emph{integral}: $x\le 1$
Let $\mathbf{A}$ and $\mathbf{B}$ be ordered monoids. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a orderpreserving homomorphism: $h(x \cdot y)=h(x) \cdot h(y)$, $h(1)=1$, $x\le y\Longrightarrow h(x)\le h(y)$.
Example 1:
$f(n)=$ number of members of size $n$.
$\begin{array}{lr}
f(1)= &1
f(2)= &1
f(3)= &2
f(4)= &8
f(5)= &44
f(6)= &308
f(7)= &2641
f(8)= &27120
f(9)= &
\end{array}$