zero elements

Let $S$ be a semigroup. An element $z$ is called a right zero [resp. left zero] if $xz=z$ [resp. $zx=z$] for all $x\in S$.

An element which is both a left and a right zero is called a zero element.

A semigroup may have many left zeros or right zeros, but if it has at least one of each, then they are necessarily equal, giving a unique (two-sided) zero element.

More generally, these definitions and statements are valid for a groupoid.

It is customary to use the symbol $\theta$ for the zero element of a semigroup.

Proposition 1.

If a groupoid has a left zero ${\mathrm{0}}_L$ and a right zero ${\mathrm{0}}_R$, then ${\mathrm{0}}_L\mathrm{=}{\mathrm{0}}_R$.

Proof.

$0_L=0_L{0}_R=0_R$. ∎

Proposition 2.

If $\mathrm{0}$ is a left zero in a semigroup $S$, then so is $x\mathit{}\mathrm{0}$ for every $x\mathrm{\in }S$.

Proof.

For any $y\in S$, $(x0)y=x(0y)=x0$. As a result, $x0$ is a left zero of $S$. ∎

Proposition 3.

If $\mathrm{0}$ is the unique left zero in a semigroup $S$, then it is also the zero element.

Proof.

By assumption and the previous proposition, $x0$ is a left zero for every $x\in S$. But $0$ is the unique left zero in $S$, we must have $x0=0$, which means that $0$ is a right zero element, and hence a zero element by the first proposition. ∎

Title	zero elements
Canonical name	ZeroElements
Date of creation	2013-03-22 13:02:19
Last modified on	2013-03-22 13:02:19
Owner	mclase (549)
Last modified by	mclase (549)
Numerical id	6
Author	mclase (549)
Entry type	Definition
Classification	msc 20N02
Classification	msc 20M99
Related topic	Semigroup
Related topic	NullSemigroup
Related topic	AbsorbingElement
Defines	zero
Defines	zero element
Defines	right zero
Defines	left zero