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A [[Partially ordered group|partially ordered group]] which is imbeddable in a complete direct product of totally ordered groups (cf. [[Totally ordered group|Totally ordered group]]). A group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096470/v0964701.png" /> is a vector group if and only if its partial order is an intersection of total orders on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096470/v0964702.png" />. A partially ordered group will be a vector group if and only if its semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096470/v0964703.png" /> of positive elements satisfies the following condition: For any finite collection of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096470/v0964704.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096470/v0964705.png" />,
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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/v/v096/v096470/v0964706.png" /></td> </tr></table>
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where this intersection is taken over all combinations of signs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096470/v0964707.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096470/v0964708.png" /> denotes the smallest invariant sub-semi-group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096470/v0964709.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096470/v09647010.png" />. An [[Orderable group|orderable group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096470/v09647011.png" /> is a vector group if and only if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096470/v09647012.png" /> it follows from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096470/v09647013.png" /> that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096470/v09647014.png" />.
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A [[Partially ordered group|partially ordered group]] which is imbeddable in a complete direct product of totally ordered groups (cf. [[Totally ordered group|Totally ordered group]]). A group  $  G $
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is a vector group if and only if its partial order is an intersection of total orders on  $  G $.
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A partially ordered group will be a vector group if and only if its semi-group  $  P $
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of positive elements satisfies the following condition: For any finite collection of elements  $  a _ {1} \dots a _ {n} $
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of  $  G $,
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$$
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\cap PS ( a _ {1} ^ {\epsilon _ {1} } \dots
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a _ {n} ^ {\epsilon _ {n} } , e )  =  P,
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$$
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where this intersection is taken over all combinations of signs $  \epsilon _ {i} = \pm  1 $,  
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while $  S ( x \dots z ) $
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denotes the smallest invariant sub-semi-group of $  G $
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containing $  x \dots z $.  
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An [[Orderable group|orderable group]] $  G $
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is a vector group if and only if for any $  g, g _ {1} \dots g _ {n} \in G $
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it follows from $  gg _ {1}  ^ {-} 1 gg _ {1} \dots g _ {n}  ^ {-} 1 gg _ {n} \in P $
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that $  g \in P $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Fuchs,  "Partially ordered algebraic systems" , Pergamon  (1963)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Fuchs,  "Partially ordered algebraic systems" , Pergamon  (1963)</TD></TR></table>

Latest revision as of 08:28, 6 June 2020


A partially ordered group which is imbeddable in a complete direct product of totally ordered groups (cf. Totally ordered group). A group $ G $ is a vector group if and only if its partial order is an intersection of total orders on $ G $. A partially ordered group will be a vector group if and only if its semi-group $ P $ of positive elements satisfies the following condition: For any finite collection of elements $ a _ {1} \dots a _ {n} $ of $ G $,

$$ \cap PS ( a _ {1} ^ {\epsilon _ {1} } \dots a _ {n} ^ {\epsilon _ {n} } , e ) = P, $$

where this intersection is taken over all combinations of signs $ \epsilon _ {i} = \pm 1 $, while $ S ( x \dots z ) $ denotes the smallest invariant sub-semi-group of $ G $ containing $ x \dots z $. An orderable group $ G $ is a vector group if and only if for any $ g, g _ {1} \dots g _ {n} \in G $ it follows from $ gg _ {1} ^ {-} 1 gg _ {1} \dots g _ {n} ^ {-} 1 gg _ {n} \in P $ that $ g \in P $.

References

[1] L. Fuchs, "Partially ordered algebraic systems" , Pergamon (1963)
How to Cite This Entry:
Vector group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vector_group&oldid=15917
This article was adapted from an original article by A.I. KokorinV.M. Kopytov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article