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#### Abbreviation: CSgrp

### Definition

A *commutative semigroup* is a semigroup *S* = (*S*,·) such that
· is commutative: *x**y* = *y**x*.

### Definition

A *commutative semigroup* is a structure *S* = (*S*,·), where · is an infix binary operation, called
the *semigroup product*, such that

· is associative: (*x**y*)*z* = *x*(*y**z*)and

· is commutative: *x**y* = *y**x*.

### Morphisms

Let *S* and *T* be commutative semigroups. A morphism from
*S* to *T* is a function *h* : *S*→*T* that is a
homomorphism:
*h*(*x**y*) = *h*(*x*)*h*(*y*).

### Some results

### Examples

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

### Properties

### 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]?