Abbreviation: CanCMon
A \emph{cancellative commutative monoid} is a cancellative monoid $\mathbf{M}=\langle M,\cdot ,e\rangle $ such that
$\cdot $ is commutative: $x\cdot y=y\cdot x$
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:
$h(x\cdot y)=h(x)\cdot h(y)$, $h(e)=e$
Example 1: $\langle\mathbb{N},+,0\rangle$, the natural numbers, with addition and zero.
All commutative free monoids are cancellative.
All finite commutative (left or right) cancellative monoids are reducts of abelian groups.
$\begin{array}{lr}
f(1)= &1
f(2)= &1
f(3)= &1
f(4)= &2
f(5)= &1
f(6)= &1
f(7)= &1
\end{array}$