# 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.
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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 |