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Difference between revisions of "Pre-orderable group"

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A [[Group|group]] for which any [[Partial order|partial order]] on it may be extended to a total order (cf. [[Orderable group|Orderable group]]). A pre-orderable group is also called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074330/p0743302.png" />-group. A group is or is not a pre-orderable group in accordance with the following criterion. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074330/p0743303.png" /> be the minimal invariant sub-semi-group of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074330/p0743304.png" /> containing an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074330/p0743305.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074330/p0743306.png" /> will then be pre-orderable if and only if, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074330/p0743307.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074330/p0743308.png" /> does not contain the unit of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074330/p0743309.png" /> and if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074330/p07433010.png" /> the intersection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074330/p07433011.png" /> is non-empty.
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A [[Group|group]] for which any [[Partial order|partial order]] on it may be extended to a total order (cf. [[Orderable group|Orderable group]]). A pre-orderable group is also called an $O^*$-group. A group is or is not a pre-orderable group in accordance with the following criterion. Let $S(g)$ be the minimal invariant sub-semi-group of a group $G$ containing an element $g$. $G$ will then be pre-orderable if and only if, for any $g\neq e$, $S(g)$ does not contain the unit of $G$ and if for any $x,y\in S(g)$ the intersection $S(x)\cap S(y)$ is non-empty.
  
All torsion-free nilpotent groups, as well as all orderable two-step solvable groups, are pre-orderable. Free groups of rank higher than 2 and free solvable groups of a solvability class higher than 2 are examples of orderable groups which are not pre-orderable. The local theorem (cf. [[Mal'tsev local theorems|Mal'tsev local theorems]]) applies to pre-orderable groups, i.e. if all finitely-generated subgroups of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074330/p07433012.png" /> are pre-orderable, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074330/p07433013.png" /> will be pre-orderable as well. However, a subgroup of a pre-orderable group need not be pre-orderable. If a quotient group of a pre-orderable group is orderable, it is pre-orderable. There exist orderable groups which are not pre-orderable, but whose quotient groups by their centres are pre-orderable. The class of pre-orderable groups is closed with respect to direct products but not with respect to complete direct products, and is consequently non-axiomatizable (cf. [[Axiomatized class|Axiomatized class]]). A wreath product of pre-orderable groups is not always pre-orderable. A subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074330/p07433014.png" /> of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074330/p07433015.png" /> is said to be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074330/p07433017.png" />-pre-orderable group if any maximal partial order on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074330/p07433018.png" /> induces a total order on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074330/p07433019.png" />.
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All torsion-free nilpotent groups, as well as all orderable two-step solvable groups, are pre-orderable. Free groups of rank higher than 2 and free solvable groups of a solvability class higher than 2 are examples of orderable groups which are not pre-orderable. The local theorem (cf. [[Mal'tsev local theorems|Mal'tsev local theorems]]) applies to pre-orderable groups, i.e. if all finitely-generated subgroups of a group $G$ are pre-orderable, $G$ will be pre-orderable as well. However, a subgroup of a pre-orderable group need not be pre-orderable. If a quotient group of a pre-orderable group is orderable, it is pre-orderable. There exist orderable groups which are not pre-orderable, but whose quotient groups by their centres are pre-orderable. The class of pre-orderable groups is closed with respect to direct products but not with respect to complete direct products, and is consequently non-axiomatizable (cf. [[Axiomatized class|Axiomatized class]]). A wreath product of pre-orderable groups is not always pre-orderable. A subgroup $H$ of a group $G$ is said to be a $\Gamma$-pre-orderable group if any maximal partial order on $G$ induces a total order on $H$.
  
 
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Latest revision as of 20:04, 12 July 2014

A group for which any partial order on it may be extended to a total order (cf. Orderable group). A pre-orderable group is also called an $O^*$-group. A group is or is not a pre-orderable group in accordance with the following criterion. Let $S(g)$ be the minimal invariant sub-semi-group of a group $G$ containing an element $g$. $G$ will then be pre-orderable if and only if, for any $g\neq e$, $S(g)$ does not contain the unit of $G$ and if for any $x,y\in S(g)$ the intersection $S(x)\cap S(y)$ is non-empty.

All torsion-free nilpotent groups, as well as all orderable two-step solvable groups, are pre-orderable. Free groups of rank higher than 2 and free solvable groups of a solvability class higher than 2 are examples of orderable groups which are not pre-orderable. The local theorem (cf. Mal'tsev local theorems) applies to pre-orderable groups, i.e. if all finitely-generated subgroups of a group $G$ are pre-orderable, $G$ will be pre-orderable as well. However, a subgroup of a pre-orderable group need not be pre-orderable. If a quotient group of a pre-orderable group is orderable, it is pre-orderable. There exist orderable groups which are not pre-orderable, but whose quotient groups by their centres are pre-orderable. The class of pre-orderable groups is closed with respect to direct products but not with respect to complete direct products, and is consequently non-axiomatizable (cf. Axiomatized class). A wreath product of pre-orderable groups is not always pre-orderable. A subgroup $H$ of a group $G$ is said to be a $\Gamma$-pre-orderable group if any maximal partial order on $G$ induces a total order on $H$.

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)


Comments

In English the term "O*-group" is generally used in preference to "pre-orderable group" ; the latter is potentially misleading, because it suggests a (non-existent) connection with the notion of a pre-order.

How to Cite This Entry:
Pre-orderable group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pre-orderable_group&oldid=13298
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