Pre-orderable group

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.