# Archimedean semi-group

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A totally ordered semi-group all strictly-positive (strictly-negative) elements of which belong to the same Archimedean class. All naturally ordered Archimedean semi-groups (cf. Naturally ordered groupoid) are isomorphic to some sub-semi-group of one of the following semi-groups: the additive semi-group of all non-negative real numbers; the semi-group of all real numbers in the interval $(0,1)$ with the usual order and with the operation $ab=\min\{a+b,1\}$; the semi-group consisting of all real numbers in the interval $(0,1)$ and the symbol $\infty$ with the usual order and with the operations:

$$ab=\begin{cases}a+b&\text{if }a+b\leq1,\\\infty&\text{if }a+b>1.\end{cases}$$

The former case occurs if and only if $S$ is a semi-group with cancellation.

How to Cite This Entry:
Archimedean semi-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Archimedean_semi-group&oldid=35781
This article was adapted from an original article by L.N. Shevrin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article