# congruence

Let $\mathrm{\Sigma}$ be a fixed signature^{}, and $\U0001d504$ a structure^{} for $\mathrm{\Sigma}$. A *congruence ^{}* $\sim $ on $\U0001d504$ is an equivalence relation

^{}such that for every natural number

^{}$n$ and $n$-ary function symbol $F$ of $\mathrm{\Sigma}$, if ${a}_{i}\sim {a}_{i}^{\prime}$ then ${F}^{\U0001d504}({a}_{1},\mathrm{\dots}{a}_{n})\sim {F}^{\U0001d504}({a}_{1}^{\prime},\mathrm{\dots}{a}_{n}^{\prime}).$

Title | congruence |
---|---|

Canonical name | Congruence12 |

Date of creation | 2013-03-22 13:46:38 |

Last modified on | 2013-03-22 13:46:38 |

Owner | almann (2526) |

Last modified by | almann (2526) |

Numerical id | 9 |

Author | almann (2526) |

Entry type | Definition |

Classification | msc 03C05 |

Classification | msc 03C07 |

Related topic | CongruenceRelationOnAnAlgebraicSystem |