# monoid

A monoid is a semigroup^{} $G$ which contains an identity element^{}; that is, there exists an element $e\in G$ such that $e\cdot a=a\cdot e=a$ for all $a\in G$.

If $e$ and $f$ are identity elements of a monoid $G$, then $e=e\cdot f=f\cdot e=f$, so we may speak of “the” identity element of $G$.

A *monoid homomorphism* from monoids $G$ to $H$ is a semigroup homomorphism $f:G\to H$ such that $f({e}_{G})={e}_{H}$, where ${e}_{G},{e}_{H}$ are identity elements of $G$ and $H$ respectively.

Title | monoid |
---|---|

Canonical name | Monoid |

Date of creation | 2013-03-22 11:50:15 |

Last modified on | 2013-03-22 11:50:15 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 9 |

Author | djao (24) |

Entry type | Definition |

Classification | msc 20M99 |

Classification | msc 34-01 |

Synonym | homomorphism^{} |

Related topic | Semigroup |

Defines | monoid homomorphism |