Abbreviation: CMon
A \emph{commutative monoid} is a monoids $\mathbf{M}=\langle M,\cdot ,e\rangle $ such that
$\cdot $ is commutative: $x\cdot y=y\cdot x$
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 $: $e\cdot x=x$
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:
$h(x\cdot y)=h(x)\cdot h(y)$, $h(e)=e$
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.
$\begin{array}{lr}
f(1)= &1
f(2)= &2
f(3)= &5
f(4)= &19
f(5)= &78
f(6)= &421
f(7)= &2637
\end{array}$