# 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 $x. 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 ArchimedeanSemigroup 2013-03-22 13:08:06 2013-03-22 13:08:06 mclase (549) mclase (549) 4 mclase (549) Definition msc 20M14 ArchimedeanProperty divides Archimedean