# 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 $0_{L}$ and a right zero $0_{R}$, then $0_{L}=0_{R}$.

###### Proof.

$0_{L}=0_{L}0_{R}=0_{R}$. ∎

###### Proposition 2.

If $0$ is a left zero in a semigroup $S$, then so is $x0$ for every $x\in S$.

###### Proof.

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

###### Proposition 3.

If $0$ 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