# identity element

Let $G$ be a groupoid^{}, that is a set with a binary operation^{} $G\times G\to G$, written muliplicatively so that $(x,y)\mapsto xy$.

An *identity element ^{}* for $G$ is an element $e$ such that $ge=eg=g$ for all $g\in G$.

The symbol $e$ is most commonly used for identity elements. Another common symbol for an identity element is $1$, particularly in semigroup^{} theory (and ring theory, considering the multiplicative structure as a semigroup).

Groups, monoids, and loops are classes of groupoids that, by definition, always have an identity element.

Title | identity element |

Canonical name | IdentityElement |

Date of creation | 2013-03-22 12:49:07 |

Last modified on | 2013-03-22 12:49:07 |

Owner | mclase (549) |

Last modified by | mclase (549) |

Numerical id | 9 |

Author | mclase (549) |

Entry type | Definition |

Classification | msc 20A05 |

Classification | msc 20N02 |

Classification | msc 20N05 |

Classification | msc 20M99 |

Synonym | neutral element |

Related topic | LeftIdentityAndRightIdentity |

Related topic | Group |