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#### Abbreviation: CanCMon

### Definition

A *cancellative commutative monoid* is a cancellative monoid *M* = (*M*,·,*e*) such that
· is commutative: *x*·*y* = *y*·*x*.

### Morphisms

Let *M* and *N* be cancellative commutative monoids. A morphism from *M*
to *N* is a function *h* : *M*→*N* that is a homomorphism:
*h*(*x*·*y*) = *h*(*x*)·*h*(*y*) and *h*(*e*) = *e*.

### Some results

All commutative free monoids are cancellative.

All finite commutative (left or right) cancellative monoids are reducts of abelian groups.

### Examples

(**N**, + ,0), the natural numbers, with addition and
zero.

### Properties

### Finite members

[Size 1]?: 1

[Size 2]?: 1

[Size 3]?: 1

[Size 4]?: 2

[Size 5]?: 1

[Size 6]?: 1

[Size 7]?: 1

### Subclasses

Abelian groups

[Cancellative commutative residuated lattices]?

### Superclasses

Cancellative commutative semigroups

Cancellative monoids

Commutative monoids