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

Finite members

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

Subclasses

Superclasses

References


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