### Table of Contents

## Cancellative commutative monoids

Abbreviation: **CanCMon**

### Definition

A \emph{cancellative commutative monoid} is a cancellative monoid $\mathbf{M}=\langle M,\cdot ,e\rangle $ such that

$\cdot $ is commutative: $x\cdot y=y\cdot x$

##### Morphisms

Let $\mathbf{M}$ and $\mathbf{N}$ be cancellative commutative monoids. A morphism from $\mathbf{M}$ to $\mathbf{N}$ is a function $h:Marrow N$ that is a homomorphism:

$h(x\cdot y)=h(x)\cdot h(y)$, $h(e)=e$

### Examples

Example 1: $\langle\mathbb{N},+,0\rangle$, the natural numbers, with addition and zero.

### Basic results

All commutative free monoids are cancellative.

All finite commutative (left or right) cancellative monoids are reducts of abelian groups.

### Properties

### Finite members

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

f(2)= &1

f(3)= &1

f(4)= &2

f(5)= &1

f(6)= &1

f(7)= &1

\end{array}$