## Commutative semigroups

Abbreviation: **CSgrp**

### Definition

A ** commutative semigroup** is a semigroups $\mathbf{S}=\langle
S,\cdot \rangle $ such that

$\cdot $ is commutative: $xy=yx$

### Definition

A ** commutative semigroup** is a structure $\mathbf{S}=\langle
S,\cdot \rangle $, where $\cdot $ is an infix binary operation, called
the

**, such that**

*semigroup product*$\cdot $ is associative: $(xy)z=x(yz)$

$\cdot $ is commutative: $xy=yx$

##### Morphisms

Let $\mathbf{S}$ and $\mathbf{T}$ be commutative semigroups. A morphism from $\mathbf{S}$ to $\mathbf{T}$ is a function $h:Sarrow T$ that is a homomorphism:

$h(xy)=h(x)h(y)$

### Examples

Example 1: $\langle \mathbb{N},+\rangle $, the natural numbers, with additition.

### Basic results

### 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 def. pr. cong. | no |

Amalgamation property | no |

Strong amalgamation property | no |

Epimorphisms are surjective | no |

### Finite members

$\begin{array}{lr} [[Search for finite commutative semigroups]] f(1)= &1\\ f(2)= &3\\ f(3)= &12\\ f(4)= &58\\ f(5)= &325\\ f(6)= &2143\\ f(7)= &17291\\ \end{array}$

### Subclasses

### Superclasses

### References

Trace: » commutative_semigroups