### Table of Contents

## Semigroups

Abbreviation: **Sgrp**

### Definition

A \emph{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: $(xy)z=x(yz)$.

##### Morphisms

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

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

### Examples

Example 1: $\langle X^{X},\circ \rangle $, the collection of functions on a sets $X$, with composition.

Example 1: $\langle \Sigma ^{+},\cdot \rangle $, the collection of nonempty strings over $\Sigma $, with concatenation.

### Basic results

### Properties

Classtype | variety |
---|---|

Equational theory | decidable in polynomial time |

Quasiequational theory | undecidable |

First-order theory | undecidable |

Locally finite | no |

Residual size | unbounded |

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}
f(1)= &1

f(2)= &5

f(3)= &24

f(4)= &188

f(5)= &1915

f(6)= &28634

f(7)= &1627672

f(8)= &3684030417

f(9)= &105\,978\,177\,936\,292

\end{array}$

[http://oeis.org/A027851 Semigroups in the Encyclopedia of Integer Sequences]