Abbreviation: OMon

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

$\le$ is 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