Cancellative monoids

A cancellative monoid is a monoid M = (M,·,e) such that · is left cancellative: z·x = z·y ⇒ x = y and
· is right cancellative: x·z = y·z ⇒ x = y.

Let M and N be cancellative 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.

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

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

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

[Size 1]?: 1
[Size 2]?: 1
[Size 3]?: 1
[Size 4]?: 2
[Size 5]?: 1
[Size 6]?: 2
[Size 7]?: 1