Abbreviation: IOMon

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

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 $n$.

$\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &2\\ f(4)= &8\\ f(5)= &44\\ f(6)= &308\\ f(7)= &2641\\ f(8)= &27120\\ f(9)= &\\ \end{array}$

Commutative integral ordered monoids

Ordered monoids