# Archimedean class

A class resulting from the subdivision induced by the Archimedean equivalence relation on a totally ordered semi-group. This equivalence is defined as follows. Two elements and of a semi-group are called Archimedean equivalent if one of the following four relations is satisfied:

which amounts to saying that and generate the same convex sub-semi-group in . Thus, the subdivision into Archimedean classes is a subdivision into pairwise non-intersecting convex sub-semi-groups. Moreover, each subdivision into pairwise non-intersecting convex sub-semi-groups, can be extended to a subdivision into Archimedean classes.

The Archimedean equivalence on a totally ordered group is induced by the Archimedean equivalence of its positive cone: It is considered that if there exist positive integers and such that

where

The positive cone of an Archimedean group consists of a single Archimedean class.

**How to Cite This Entry:**

Archimedean class. O.A. Ivanova (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Archimedean_class&oldid=11365