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.

$\begin{array}{lr} f(1)= &1 f(2)= &1 f(3)= &1 f(4)= &2 f(5)= &1 f(6)= &1 f(7)= &1 \end{array}$

Abelian groups

Cancellative commutative residuated lattices

Cancellative commutative semigroups

Cancellative monoids

Commutative monoids