# congruence

Let $S$ be a semigroup^{}. An equivalence relation^{} $\sim $ defined on $S$ is called a *congruence ^{}* if it is preserved under the semigroup operation

^{}. That is, for all $x,y,z\in S$, if $x\sim y$ then $xz\sim yz$ and $zx\sim zy$.

If $\sim $ satisfies only $x\sim y$ implies $xz\sim yz$ (resp. $zx\sim zy$) then $\sim $ is called a *right congruence* (resp. *left congruence*).

###### Example.

Suppose $f\mathrm{:}S\mathrm{\to}T$ is a semigroup homomorphism. Define $\mathrm{\sim}$ by $x\mathrm{\sim}y$ iff $f\mathit{}\mathrm{(}x\mathrm{)}\mathrm{=}f\mathit{}\mathrm{(}y\mathrm{)}$. Then it is easy to see that $\mathrm{\sim}$ is a congruence.

If $\sim $ is a congruence, defined on a semigroup $S$,
write $[x]$ for the equivalence class^{} of $x$ under $\sim $.
Then it is easy to see that $[x]\cdot [y]=[xy]$
is a well-defined operation on the set of equivalence classes,
and that in fact this set becomes a semigroup with this operation.
This semigroup is called the *quotient of $S$ by $\mathrm{\sim}$*
and is written $S/\sim $.

Thus semigroup are related to homomorphic images^{} of semigroups in the same way that normal subgroups^{} are related to homomorphic images of groups. More precisely, in the group case, the congruence is the coset relation^{}, rather than the normal subgroup itself.

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

Canonical name | Congruence1 |

Date of creation | 2013-03-22 13:01:08 |

Last modified on | 2013-03-22 13:01:08 |

Owner | mclase (549) |

Last modified by | mclase (549) |

Numerical id | 7 |

Author | mclase (549) |

Entry type | Definition |

Classification | msc 20M99 |

Related topic | Congruences |

Related topic | MultiplicativeCongruence |

Related topic | CongruenceRelationOnAnAlgebraicSystem |

Defines | quotient semigroup |