### Table of Contents

## Commutative monoids

Abbreviation: **CMon**

### Definition

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

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

### Definition

A \emph{commutative monoid} is a structure $\mathbf{M}=\langle M,\cdot ,e\rangle $, where $\cdot $ is an infix binary operation, called the \emph{monoid product}, and $e$ is a constant (nullary operation), called the \emph{identity element}, such that

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

$\cdot $ is associative: $(x\cdot y)\cdot z=x\cdot (y\cdot z)$

$e$ is an identity for $\cdot $: $e\cdot x=x$

##### Morphisms

Let $\mathbf{M}$ and $\mathbf{N}$ be commutative monoids. A morphism from $\mathbf{M}$ to $\mathbf{N}$ is a function $h:M\to N$ 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. The finitely generated free commutative monoids are direct products of this one.

### Basic results

### Properties

### Finite members

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

f(2)= &2

f(3)= &5

f(4)= &19

f(5)= &78

f(6)= &421

f(7)= &2637

\end{array}$