# Right-ordered group

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. ). 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).

How to Cite This Entry:
Right-ordered group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Right-ordered_group&oldid=43497
This article was adapted from an original article by V.M. Kopytov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article