Abbreviation: Sgrp$_0$

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 $\cdot$: $x\cdot 0=0$, $0\cdot x=0$

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:

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

Example 1:

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

Commutative semigroups with zero

Semigroups