Abbreviation: OMon

An \emph{ordered monoid} is a partially ordered monoid $\mathbf{A}=\langle A,\cdot,1,\le\rangle$ such that

$\le$ is \emph{linear}: $x\le y\text{ or }y\le x$

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 $n$.

$\begin{array}{lr} f(1)= &1
f(2)= &2
f(3)= &8
f(4)= &34
f(5)= &184
f(6)= &1218
f(7)= &9742
f(8)= &
f(9)= &
\end{array}$

Commutative ordered monoids

Partially ordered monoids

Ordered semigroups reduced type