### Table of Contents

## Semigroups with zero

Abbreviation: **Sgrp$_0$**

### Definition

A \emph{semigroup with zero} is a structure $\mathbf{S}=\langle S,\cdot,0\rangle$ of type $\langle 2,0\rangle $ such that

$\langle S,\cdot\rangle$ is a semigroups

$0$ is a zero for $\cdot$: $x\cdot 0=0$, $0\cdot x=0$

##### Morphisms

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

$h(x\cdot y)=h(x)\cdot h(y)$, $h(0)=0$

### Examples

Example 1:

### Basic results

### Properties

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

Equational theory | decidable in PTIME |

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. | |

Amalgamation property | |

Strong amalgamation property | |

Epimorphisms are surjective |

### Finite members

$\begin{array}{lr}
f(1)= &1

f(2)= &

f(3)= &

f(4)= &

f(5)= &

f(6)= &

\end{array}$