zero elementsZeroElements2002-09-07 11:56:462008-12-08 15:00:12Definitionzerozero elementright zeroleft zero% this is the default PlanetMath preamble. as your knowledge
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\newtheorem{prop}{Proposition}Let $S$ be a semigroup. An element $z$ is called a \emph{right zero} [resp. \emph{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 \emph{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.
\begin{prop} If a groupoid has a left zero $0_L$ and a right zero $0_R$, then $0_L = 0_R$. \end{prop}
\begin{proof} $0_L=0_L 0_R = 0_R$. \end{proof}
\begin{prop} If $0$ is a left zero in a semigroup $S$, then so is $x0$ for every $x\in S$. \end{prop}
\begin{proof}
For any $y\in S$, $(x0)y=x(0y)=x0$. As a result, $x0$ is a left zero of $S$.
\end{proof}
\begin{prop} If $0$ is the unique left zero in a semigroup $S$, then it is also the zero element. \end{prop}
\begin{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.
\end{proof}