# Commutative semigroups

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### Definition

A commutative semigroup is a semigroup S = (S,·) such that · is commutative:   xy = yx.

### Definition

A commutative semigroup is a structure S = (S,·), where · is an infix binary operation, called the semigroup product, such that
· is associative:   (xy)z = x(yz)and
· is commutative:   xy = yx.

### Morphisms

Let S and T be commutative 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 variety Equational theory decidable in polynomial time Quasiequational theory decidable 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

### Finite members

Search for finite commutative semigroups

Size 1:  1
Size 2:  3
Size 3:  12
Size 4:  58
Size 5:  325
[Size 6]?:  2143
[Size 7]?:  17291

### Subclasses

Semilattices
Commutative monoids

### Superclasses

Semigroups
[Partial commutative semigroups]?

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