## You are here

HomeArchimedean semigroup

## Primary tabs

# 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.

Defines:

divides, Archimedean

Related:

ArchimedeanProperty

Type of Math Object:

Definition

Major Section:

Reference

## Mathematics Subject Classification

20M14*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

Oct 21

new question: Prime numbers out of sequence by Rubens373

Oct 7

new question: Lorenz system by David Bankom

Oct 19

new correction: examples and OEIS sequences by fizzie

Oct 13

new correction: Define Galois correspondence by porton

Oct 7

new correction: Closure properties on languages: DCFL not closed under reversal by babou

new correction: DCFLs are not closed under reversal by petey

Oct 2

new correction: Many corrections by Smarandache

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag

new question: Prime numbers out of sequence by Rubens373

Oct 7

new question: Lorenz system by David Bankom

Oct 19

new correction: examples and OEIS sequences by fizzie

Oct 13

new correction: Define Galois correspondence by porton

Oct 7

new correction: Closure properties on languages: DCFL not closed under reversal by babou

new correction: DCFLs are not closed under reversal by petey

Oct 2

new correction: Many corrections by Smarandache

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag