### Table of Contents

## Ordered semigroups

Abbreviation: **OSgrp**

### Definition

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

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

##### Morphisms

Let $\mathbf{A}$ and $\mathbf{B}$ be ordered semigroups. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a orderpreserving homomorphism: $h(x \cdot y)=h(x) \cdot h(y)$, $x\le y\Longrightarrow h(x)\le h(y)$.

### Examples

Example 1:

### Basic results

### Properties

### Finite members

$\begin{array}{rr}
f(1)=&1

f(2)=&6

f(3)=&44

f(4)=&386

f(5)=&3852

f(6)=&42640

f(7)=&516791

f(8)=&6817378

\end{array}$

### Subclasses

### Superclasses

Chains reduced type