Difference between revisions of "Archimedean class"
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| − | + | A class resulting from the subdivision induced by the Archimedean equivalence relation on a totally [[Ordered semi-group|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|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 | where | ||
| − | + | $$ | |
| + | | x | = \max \{ x , x ^ {-1} \} . | ||
| + | $$ | ||
The positive cone of an [[Archimedean group|Archimedean group]] consists of a single Archimedean class. | The positive cone of an [[Archimedean group|Archimedean group]] consists of a single Archimedean class. | ||
Latest revision as of 18:48, 5 April 2020
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.
Archimedean class. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Archimedean_class&oldid=11365