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A group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070100/o0701001.png" /> on which a total order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070100/o0701002.png" /> (cf. [[Totally ordered group|Totally ordered group]]) can be introduced such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070100/o0701003.png" /> entails <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070100/o0701004.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070100/o0701005.png" />. A group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070100/o0701006.png" /> is orderable as a group if and only if there exists a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070100/o0701007.png" /> with the properties: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070100/o0701008.png" />; 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070100/o0701009.png" />, 3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070100/o07010010.png" />; 4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070100/o07010011.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070100/o07010012.png" />.
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$#C+1 = 23 : ~/encyclopedia/old_files/data/O070/O.0700100 Orderable group
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070100/o07010013.png" /> be the normal sub-semi-group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070100/o07010014.png" /> generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070100/o07010015.png" />. The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070100/o07010016.png" /> is orderable as a group if and only if for any finite set of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070100/o07010017.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070100/o07010018.png" />, different from the unit element, numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070100/o07010019.png" /> can be found, equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070100/o07010020.png" />, such that the sub-semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070100/o07010021.png" /> does not contain the unit element. Every orderable group is a group with unique root extraction. Abelian torsion-free groups, locally nilpotent torsion-free groups, free, and free solvable groups are orderable groups. Two-step solvable groups such that for every non-unit element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070100/o07010022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070100/o07010023.png" />, are orderable groups.
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A group  $  G $
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on which a total order  $  \leq  $(
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cf. [[Totally ordered group|Totally ordered group]]) can be introduced such that  $  a \leq  b $
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entails  $  x ay \leq  x b y $
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for any  $  a , b , x , y \in G $.
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A group  $  G $
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is orderable as a group if and only if there exists a subset  $  P $
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with the properties: 1)  $  P P \subseteq P $;
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2)  $  P \cap P  ^ {-} 1 = \{ 1 \} $,
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3)  $  P \cup P  ^ {-} 1 = G $;
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4)  $  x  ^ {-} 1 P x \subseteq P $
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for any  $  x \in G $.
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Let  $  S ( a _ {1} \dots a _ {n} ) $
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be the normal sub-semi-group of $  G $
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generated by $  a _ {1} \dots a _ {n} $.  
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The group $  G $
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is orderable as a group if and only if for any finite set of elements $  a _ {1} \dots a _ {n} $
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in $  G $,  
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different from the unit element, numbers $  \epsilon _ {1} \dots \epsilon _ {n} $
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can be found, equal to $  \pm  1 $,  
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such that the sub-semi-group $  S ( a _ {1} ^ {\epsilon _ {1} } \dots a _ {n} ^ {\epsilon _ {n} } ) $
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does not contain the unit element. Every orderable group is a group with unique root extraction. Abelian torsion-free groups, locally nilpotent torsion-free groups, free, and free solvable groups are orderable groups. Two-step solvable groups such that for every non-unit element $  x $,  
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$  1 \notin S ( x) $,  
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are orderable groups.
  
 
The class of orderable groups is closed under taking subgroups and direct products; it is locally closed, and consequently, a [[Quasi-variety|quasi-variety]]. A free product of orderable groups is again an orderable group.
 
The class of orderable groups is closed under taking subgroups and direct products; it is locally closed, and consequently, a [[Quasi-variety|quasi-variety]]. A free product of orderable groups is again an orderable group.

Latest revision as of 08:04, 6 June 2020


A group $ G $ on which a total order $ \leq $( cf. Totally ordered group) can be introduced such that $ a \leq b $ entails $ x ay \leq x b y $ for any $ a , b , x , y \in G $. A group $ G $ is orderable as a group if and only if there exists a subset $ P $ with the properties: 1) $ P P \subseteq P $; 2) $ P \cap P ^ {-} 1 = \{ 1 \} $, 3) $ P \cup P ^ {-} 1 = G $; 4) $ x ^ {-} 1 P x \subseteq P $ for any $ x \in G $.

Let $ S ( a _ {1} \dots a _ {n} ) $ be the normal sub-semi-group of $ G $ generated by $ a _ {1} \dots a _ {n} $. The group $ G $ is orderable as a group if and only if for any finite set of elements $ a _ {1} \dots a _ {n} $ in $ G $, different from the unit element, numbers $ \epsilon _ {1} \dots \epsilon _ {n} $ can be found, equal to $ \pm 1 $, such that the sub-semi-group $ S ( a _ {1} ^ {\epsilon _ {1} } \dots a _ {n} ^ {\epsilon _ {n} } ) $ does not contain the unit element. Every orderable group is a group with unique root extraction. Abelian torsion-free groups, locally nilpotent torsion-free groups, free, and free solvable groups are orderable groups. Two-step solvable groups such that for every non-unit element $ x $, $ 1 \notin S ( x) $, are orderable groups.

The class of orderable groups is closed under taking subgroups and direct products; it is locally closed, and consequently, a quasi-variety. A free product of orderable groups is again an orderable group.

References

[1] A.I. Kokorin, V.M. Kopytov, "Fully ordered groups" , Israel Program Sci. Transl. (1974) (Translated from Russian)
[2] L. Fuchs, "Partially ordered algebraic systems" , Pergamon (1963)
How to Cite This Entry:
Orderable group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Orderable_group&oldid=48067
This article was adapted from an original article by V.M. Kopytov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article