# Totally ordered group

An algebraic system $G$ that is a group with respect to multiplication, a totally ordered set with respect to a binary order relation $\leq$ and satisfies the following axiom: For any elements $x,y,z\in G$, from $x\leq y$ it follows that $xz\leq yz$ and $zx\leq zy$.

The set of positive elements $P=\{x\in G\colon x\geq e\}$ of a totally ordered group $G$ has the following properties: 1) $PP\subseteq P$; 2) $P\cap P^{-1}=e$; 3) $g^{-1}Pg\subseteq P$; and 4) $P\cup P^{-1}=G$. Conversely, if in a group $G$ there is a set $P$ satisfying conditions 1)–4), then $G$ can be made into a totally ordered group with $P$ as set of positive elements.

There is a large number of criteria for a group to be orderable. Orderable groups are torsion-free groups with unique root extraction. The following groups are orderable: torsion-free Abelian groups, torsion-free nilpotent groups, free groups, and free solvable groups. Simple non-Hopfian totally ordered groups exist. The quotient group of an orderable group by its centre is orderable.

The direct product, the complete direct product and the free product, and also the wreath product, of totally ordered groups can be totally ordered by extending the orders of the factors. A group that can be approximated by orderable groups is itself orderable. For orderable groups there is a local theorem (see Mal'tsev local theorems). A totally ordered group can be imbedded in the multiplicative group of a totally ordered skew-field and in a simple totally ordered group. The class of orderable groups is axiomatizable. A totally ordered group is a topological group with respect to the interval topology. A totally ordered group is called Archimedean if and only if it does not have non-trivial convex subgroups. Any Archimedean totally ordered group is isomorphic to a subgroup of the additive group of real numbers with the natural order. The set of all convex subgroups of a totally ordered group forms a complete infra-invariant system with Archimedean factors, and so totally ordered groups have solvable normal systems (see Subgroup system).

Specific for the theory of totally ordered groups are questions connected with the extension of partial orders (see Pre-orderable group). There is a number of generalizations of the concept of a totally ordered group.

#### 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) [3] N. Bourbaki, "Algèbre" , Eléments de mathématiques , Masson (1981) pp. Chapts. IV-VI