Cancellative commutative monoids

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

Classtype	quasivariety
Equational theory
Quasiequational theory
First-order theory	undecidable
Locally finite	no
Residual size	unbounded
Congruence distributive	no
Congruence modular
Congruence n-permutable
Congruence regular
Congruence uniform
Congruence extension property
Definable principal congruences
Equationally definable principal congruences
Amalgamation property
Strong amalgamation property
Epimorphisms are surjective

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