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

Classtype quasivariety undecidable no unbounded no

Finite members

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