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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013110/a0131101.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013110/a0131102.png" /> of a semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013110/a0131103.png" /> are called Archimedean equivalent if one of the following four relations is satisfied:
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013110/a0131104.png" /></td> </tr></table>
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which amounts to saying that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013110/a0131105.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013110/a0131106.png" /> generate the same convex sub-semi-group in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013110/a0131107.png" />. 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.
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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 $
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and  $  b $
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of a semi-group  $  S $
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are called Archimedean equivalent if one of the following four relations is satisfied:
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013110/a0131108.png" /> if there exist positive integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013110/a0131109.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013110/a01311010.png" /> such that
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$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013110/a01311011.png" /></td> </tr></table>
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\begin{array}{ll}
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a \leq  b \leq  a  ^ {n} ,\  &b \leq  a \leq  b  ^ {n} ,\  \\
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a  ^ {n} \leq  b \leq  a ,  &b  ^ {n} \leq  a \leq b ; \\
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\end{array}
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$$
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which amounts to saying that  $  a $
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and  $  b $
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generate the same convex sub-semi-group in  $  S $.  
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Thus, the subdivision into Archimedean classes is a subdivision into pairwise non-intersecting convex sub-semi-groups. Moreover, each subdivision into pairwise non-intersecting convex sub-semi-groups, can be extended to a subdivision into Archimedean classes.
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The Archimedean equivalence on a [[Totally ordered group|totally ordered group]] is induced by the Archimedean equivalence of its positive cone: It is considered that  $  a \sim b $
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if there exist positive integers  $  m $
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and  $  n $
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such that
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$$
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| a |  < | b |  ^ {m} \  \textrm{ and } \ \
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| b |  < | a |  ^ {n} ,
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$$
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013110/a01311012.png" /></td> </tr></table>
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$$
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| x |  = \max \{ x , x  ^ {-1} \} .
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$$
  
 
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.

How to Cite This Entry:
Archimedean class. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Archimedean_class&oldid=11365
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article