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
$\begin{array}{lr}
f(1)= &1
f(2)= &
f(3)= &
f(4)= &
f(5)= &
f(6)= &
\end{array}$
