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.

单词	ArchimedeanSemigroup
释义	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.
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