# Archimedean semigroup

Let $S$ be a commutative semigroup. We say an element $x$ *divides* an element $y$, written $x\mid y$, if there is an element $z$ such that $xz=y$.

An *Archimedean semigroup* $S$ is a commutative semigroup with the property that for all $x,y\in S$ there is a natural number^{} $n$ such that $x\mid {y}^{n}$.

This is related to the Archimedean property of positive real numbers ${\mathbb{R}}^{+}$: if $x,y>0$ then there is a natural number $n$ such that $$. Except that the notation is additive rather than multiplicative, this is the same as saying that $({\mathbb{R}}^{+},+)$ is an Archimedean semigroup.

Title | Archimedean semigroup |
---|---|

Canonical name | ArchimedeanSemigroup |

Date of creation | 2013-03-22 13:08:06 |

Last modified on | 2013-03-22 13:08:06 |

Owner | mclase (549) |

Last modified by | mclase (549) |

Numerical id | 4 |

Author | mclase (549) |

Entry type | Definition |

Classification | msc 20M14 |

Related topic | ArchimedeanProperty |

Defines | divides |

Defines | Archimedean^{} |