# Cancellative semigroups

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### Definition

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

### Morphisms

Let S and T be cancellative semigroups. A morphism from S to T is a function h : ST that is a homomorphism: h(xy) = h(x)h(y).

### Examples

(N, + ), the natural numbers, with additition.

### Properties

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

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### Subclasses

Cancellative monoids

### Superclasses

Semigroups

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