# left identity and right identity

Let $G$ be a groupoid^{}. An element $e\in G$ is called a *left identity ^{} element* if $ex=x$ for all $x\in G$. Similarly, $e$ is a

*right identity element*if $xe=x$ for all $x\in G$.

An element which is both a left and a right identity is an identity element^{}.

A groupoid may have more than one left identify element: in fact the operation defined by $xy=y$ for all $x,y\in G$ defines a groupoid (in fact, a semigroup) on any set $G$, and every element is a left identity.

But as soon as a groupoid has both a left and a right identity, they are necessarily unique and equal. For if $e$ is a left identity and $f$ is a right identity, then $f=ef=e$.

Title | left identity and right identity |
---|---|

Canonical name | LeftIdentityAndRightIdentity |

Date of creation | 2013-03-22 13:02:05 |

Last modified on | 2013-03-22 13:02:05 |

Owner | mclase (549) |

Last modified by | mclase (549) |

Numerical id | 5 |

Author | mclase (549) |

Entry type | Definition |

Classification | msc 20N02 |

Classification | msc 20M99 |

Related topic | IdentityElement |

Related topic | Unity |

Defines | left identity |

Defines | right identity |