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 $x<ny$ . 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