Archimedean semigroupArchimedeanSemigroup2002-11-05 18:38:392002-11-05 18:38:39DefinitiondividesArchimedean% this is the default PlanetMath preamble. as your knowledge
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% define commands hereLet $S$ be a commutative semigroup. We say an element $x$ \emph{divides} an element $y$, written $x \mid y$, if there is an element $z$ such that $xz = y$.
An \emph{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.