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 $ a $ and $ b $ of a semi-group $ S $ are called Archimedean equivalent if one of the following four relations is satisfied:

$$ \begin{array}{ll} a \leq b \leq a ^ {n} ,\ &b \leq a \leq b ^ {n} ,\ \\ a ^ {n} \leq b \leq a , &b ^ {n} \leq a \leq b ; \\ \end{array} $$

which amounts to saying that $ a $ and $ b $ generate the same convex sub-semi-group in $ S $. 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 $ a \sim b $ if there exist positive integers $ m $ and $ n $ such that

$$ | a | < | b | ^ {m} \ \textrm{ and } \ \ | b | < | a | ^ {n} , $$

where

$$ | x | = \max \{ x , x ^ {-1} \} . $$

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