O-group

totally ordered group

A $ po $- group $ G $ whose partial order is total (cf. also Totally ordered group). A $ po $- group is an $ o $- group if and only if $ P ( G ) \cup P ( G ) ^ {- 1 } = G $, where $ P ( G ) $ is the positive cone of $ G $. Every Abelian torsion-free group, every locally nilpotent-torsion free group and every free group can be turned into an $ o $- group. Any $ o $- group $ G $ is a quotient group of a free $ o $- group $ F $ with suitable order by some convex normal subgroup. Direct, Cartesian, free, and direct wreath products of $ o $- groups can be turned in $ o $- groups by extending the orders of the factors. Any $ o $- group is a topological group respect with to the interval topology. Any $ o $- group can be imbedded into simple $ o $- groups. There exist non-Hopfian $ o $- groups.

If $ G $ is an $ o $- group, then the group-theoretical structure of $ G $ is very nice. In particular, any $ o $- group has a subnormal solvable system of subgroups, it is a torsion-free group and a group with unique roots. Thus, $ o $- groups are suitable examples in the study of many classes of groups. The system $ {\mathcal C} ( G ) $ of convex subgroups of an $ o $- group is well studied (cf. Convex subgroup). In particular, $ {\mathcal C} ( G ) $ is the central series of isolated normal subgroups for any locally nilpotent $ o $- group.

It is useful to apply methods from the theory of semi-groups to questions about orderability of a group. Let $ S ( X ) $ be the normal sub-semi-group of a group $ G $ generated by a set $ X \subset G $. Then $ G $ is orderable (i.e., it is possible to turn $ G $ in an $ o $- group) if and only if $ e \notin S ( a _ {1} \dots a _ {n} ) $ for all $ a _ {1} \dots a _ {n} \in G $( $ a _ {i} \neq e $). Every partial order $ Q $ can be extended to a total order on $ G $ if and only if for all $ x \in G $, $ x \neq e $ implies $ e \notin S ( x ) $ and for all $ x,y,z \in G $,

$$ x,y \in S ( z ) \Rightarrow S ( x ) \cap S ( y ) \neq \emptyset. $$

Groups with this property are called fully orderable. If a group is torsion-free and Abelian, or locally nilpotent or orderable metabelian, then it is fully orderable. The class of fully orderable groups is closed under formation of direct products and is locally closed. It is not closed under formation of subgroups, Cartesian products, free products. It is non-axiomatizable.

This article complements the article Totally ordered group (Vol. 9).

References

[a1]	L. Fuchs, "Partially ordered algebraic systems" , Pergamon (1963)
[a2]	A.I. Kokorin, V.M. Kopytov, "Fully ordered groups" , Halstad (1974) (In Russian)
[a3]	R.T.B. Mura, A.H. Rhemtulla, "Orderable groups" , M. Dekker (1977)
[a4]	V.M. Kopytov, N.Ya. Medvedev, "The theory of lattice-ordered groups" , Kluwer Acad. Publ. (1994) (In Russian)