### Table of Contents

## Commutative semigroups

Abbreviation: **CSgrp**

### Definition

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

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

### Definition

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

$\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}$