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# 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

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Reference

## Mathematics Subject Classification

20M14*no label found*

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