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 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.
Archimedean class. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Archimedean_class&oldid=45211