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. [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