### Table of Contents

## Cancellative semigroups

Abbreviation: **CanSgrp**

### Definition

A \emph{cancellative semigroup} is a semigroup $\mathbf{S}=\langle S,\cdot\rangle$ such that

$\cdot $ is left cancellative: $z\cdot x=z\cdot y\Longrightarrow x=y$

$\cdot $ is right cancellative: $x\cdot z=y\cdot z\Longrightarrow x=y$

##### Morphisms

Let $\mathbf{S}$ and $\mathbf{T}$ be cancellative semigroups. A morphism from $\mathbf{S}$ to $\mathbf{T}$ is a function $h:S\rightarrow T$ that is a homomorphism:

$h(xy)=h(x)h(y)$

### Examples

Example 1: $\langle \mathbb{N},+\rangle $, the natural numbers, with additition.

### Basic results

### Properties

### Finite members

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

f(2)= &

f(3)= &

f(4)= &

f(5)= &

f(6)= &

f(7)= &

\end{array}$