### Table of Contents

## Integral ordered monoids

Abbreviation: **IOMon**

### Definition

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

\emph{integral}: $x\le 1$

##### Morphisms

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: $h(x \cdot y)=h(x) \cdot h(y)$, $h(1)=1$, $x\le y\Longrightarrow h(x)\le h(y)$.

### Examples

Example 1:

### Basic results

### Properties

### Finite members

$f(n)=$ 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}$