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A [[Group|group]] on whose set of elements a total order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082370/r0823701.png" /> (cf. [[Totally ordered group|Totally ordered group]]) is defined such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082370/r0823702.png" /> the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082370/r0823703.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082370/r0823704.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082370/r0823705.png" /> of positive elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082370/r0823706.png" /> is a pure (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082370/r0823707.png" />) linear (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082370/r0823708.png" />) sub-semi-group. Every pure linear sub-semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082370/r0823709.png" /> of an arbitrary group defines a right order, namely <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082370/r08237010.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082370/r08237011.png" />.
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A [[Group|group]] on whose set of elements a total order $\leq$ (cf. [[Totally ordered group|Totally ordered group]]) is defined such that for all $x,y,z\in G$ the inequality $x\leq y$ implies $xz\leq yz$. The set $P=\{x\in G:x>e\}$ of positive elements of $G$ is a pure (i.e. $P\cap P^{-1}=\emptyset$) linear (i.e. $P\cup P^{-1}\cup\{e\}=G$) sub-semi-group. Every pure linear sub-semi-group $P$ of an arbitrary group defines a right order, namely $x<y$ if and only if $yx^{-1}\in P$.
  
The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082370/r08237012.png" /> of automorphisms of a totally-ordered set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082370/r08237013.png" /> can be right ordered in a natural manner. Every right-ordered group is order-isomorphic to some subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082370/r08237014.png" /> for a suitable totally-ordered set (cf. [[#References|[1]]]). An Archimedean right-ordered group, i.e. a right-ordered group for which Archimedes' axiom holds (cf. [[Archimedean group|Archimedean group]]), is order-isomorphic to a subgroup of the additive group of real numbers. In contrast with (two-sided) ordered groups, there are non-commutative right-ordered groups without proper convex subgroups (cf. [[Convex subgroup|Convex subgroup]]). The class of right-ordered groups is closed under lexicographic extension. The system of all convex subgroups of a right-ordered group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082370/r08237015.png" /> is totally ordered with respect to inclusion and is complete. This system is solvable (cf. also [[Solvable group|Solvable group]]) if and only if for any positive elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082370/r08237016.png" /> there is a natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082370/r08237017.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082370/r08237018.png" />. If the group has a solvable subgroup system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082370/r08237019.png" /> whose factors are torsion-free, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082370/r08237020.png" /> can be right-ordered in such a way that all subgroups in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082370/r08237021.png" /> become convex. In a locally nilpotent right-ordered group the system of convex subgroups is solvable.
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The group $A(X)$ of automorphisms of a totally-ordered set $X$ can be right ordered in a natural manner. Every right-ordered group is order-isomorphic to some subgroup of $A(X)$ for a suitable totally-ordered set (cf. [[#References|[1]]]). An Archimedean right-ordered group, i.e. a right-ordered group for which Archimedes' axiom holds (cf. [[Archimedean group|Archimedean group]]), is order-isomorphic to a subgroup of the additive group of real numbers. In contrast with (two-sided) ordered groups, there are non-commutative right-ordered groups without proper convex subgroups (cf. [[Convex subgroup|Convex subgroup]]). The class of right-ordered groups is closed under lexicographic extension. The system of all convex subgroups of a right-ordered group $G$ is totally ordered with respect to inclusion and is complete. This system is solvable (cf. also [[Solvable group|Solvable group]]) if and only if for any positive elements $a,b\in G$ there is a natural number $n$ such that $a^nb>a$. If the group has a solvable subgroup system $S(G)$ whose factors are torsion-free, then $G$ can be right-ordered in such a way that all subgroups in $S(G)$ become convex. In a locally nilpotent right-ordered group the system of convex subgroups is solvable.
  
A group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082370/r08237022.png" /> can be right-ordered if and only if for any finite system
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A group $G$ can be right-ordered if and only if for any finite system
  
<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/r/r082/r082370/r08237023.png" /></td> </tr></table>
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$$\{x_i\neq e:1\leq i\leq n\}$$
  
of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082370/r08237024.png" /> there are numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082370/r08237025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082370/r08237026.png" />, such that the semi-group generated by the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082370/r08237027.png" /> does not contain the identity element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082370/r08237028.png" />.
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of elements of $G$ there are numbers $\epsilon_i=\pm1$, $1\leq i\leq n$, such that the semi-group generated by the set $\{x_1^{\epsilon_1},\dots,x_n^{\epsilon_n}\}$ does not contain the identity element of $G$.
  
Every lattice ordering of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082370/r08237029.png" /> is the intersection of some of its right-orderings (cf. [[Lattice-ordered group|Lattice-ordered group]]).
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Every lattice ordering of a group $G$ is the intersection of some of its right-orderings (cf. [[Lattice-ordered group|Lattice-ordered group]]).
  
 
====References====
 
====References====
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====Comments====
 
====Comments====
A group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082370/r08237030.png" /> that admits a total order such that with this order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082370/r08237031.png" /> becomes a right-ordered group, is called right-orderable. Such an order on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082370/r08237032.png" /> is called a right order or right ordering.
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A group $G$ that admits a total order such that with this order $G$ becomes a right-ordered group, is called right-orderable. Such an order on $G$ is called a right order or right ordering.
  
 
Some more concepts and results concerning right-ordered groups can be found in [[#References|[a1]]]–[[#References|[a4]]].
 
Some more concepts and results concerning right-ordered groups can be found in [[#References|[a1]]]–[[#References|[a4]]].

Latest revision as of 11:03, 27 November 2018

A group on whose set of elements a total order $\leq$ (cf. Totally ordered group) is defined such that for all $x,y,z\in G$ the inequality $x\leq y$ implies $xz\leq yz$. The set $P=\{x\in G:x>e\}$ of positive elements of $G$ is a pure (i.e. $P\cap P^{-1}=\emptyset$) linear (i.e. $P\cup P^{-1}\cup\{e\}=G$) sub-semi-group. Every pure linear sub-semi-group $P$ of an arbitrary group defines a right order, namely $x<y$ if and only if $yx^{-1}\in P$.

The group $A(X)$ of automorphisms of a totally-ordered set $X$ can be right ordered in a natural manner. Every right-ordered group is order-isomorphic to some subgroup of $A(X)$ for a suitable totally-ordered set (cf. [1]). An Archimedean right-ordered group, i.e. a right-ordered group for which Archimedes' axiom holds (cf. Archimedean group), is order-isomorphic to a subgroup of the additive group of real numbers. In contrast with (two-sided) ordered groups, there are non-commutative right-ordered groups without proper convex subgroups (cf. Convex subgroup). The class of right-ordered groups is closed under lexicographic extension. The system of all convex subgroups of a right-ordered group $G$ is totally ordered with respect to inclusion and is complete. This system is solvable (cf. also Solvable group) if and only if for any positive elements $a,b\in G$ there is a natural number $n$ such that $a^nb>a$. If the group has a solvable subgroup system $S(G)$ whose factors are torsion-free, then $G$ can be right-ordered in such a way that all subgroups in $S(G)$ become convex. In a locally nilpotent right-ordered group the system of convex subgroups is solvable.

A group $G$ can be right-ordered if and only if for any finite system

$$\{x_i\neq e:1\leq i\leq n\}$$

of elements of $G$ there are numbers $\epsilon_i=\pm1$, $1\leq i\leq n$, such that the semi-group generated by the set $\{x_1^{\epsilon_1},\dots,x_n^{\epsilon_n}\}$ does not contain the identity element of $G$.

Every lattice ordering of a group $G$ is the intersection of some of its right-orderings (cf. Lattice-ordered group).

References

[1] A.I. Kokorin, V.M. Kopytov, "Fully ordered groups" , Israel Program Sci. Transl. (1974) (Translated from Russian)
[2] R.B. Mura, A. Rhemtulla, "Orderable groups" , M. Dekker (1977)


Comments

A group $G$ that admits a total order such that with this order $G$ becomes a right-ordered group, is called right-orderable. Such an order on $G$ is called a right order or right ordering.

Some more concepts and results concerning right-ordered groups can be found in [a1][a4].

References

[a1] M. Anderson, T. Feil, "Lattice-ordered groups. An introduction" , Reidel (1988) pp. 35; 38ff
[a2] A.M.W. Glass (ed.) W.Ch. Holland (ed.) , Lattice-ordered groups. Advances and techniques , Kluwer (1989)
[a3] W.B. Powell, "Universal aspects of the theory of lattice-ordered groups" J. Martinez (ed.) , Ordered Algebraic Structures , Kluwer (1989) pp. 11–50
[a4] M.R. Darnell, "Recent results on the free lattice ordered group over a right-orderable group" J. Martinez (ed.) , Ordered Algebraic Structures , Kluwer (1989) pp. 51–57
How to Cite This Entry:
Right-ordered group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Right-ordered_group&oldid=15087
This article was adapted from an original article by V.M. Kopytov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article