### Table of Contents

## Ordered monoids

Abbreviation: **OMon**

### Definition

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

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

##### 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)= &2

f(3)= &8

f(4)= &34

f(5)= &184

f(6)= &1218

f(7)= &9742

f(8)= &

f(9)= &

\end{array}$

### Subclasses

### Superclasses

Ordered semigroups reduced type