Ordered group

A group with an order relation such that for any in the inequality entails . If the order is total (respectively, partial), one speaks of a totally ordered group (respectively, a partially ordered group).

An order homomorphism of a (partially) ordered group into an ordered group is a homomorphism of into such that , , implies in . The kernels of order homomorphisms are the convex normal subgroups (cf. Convex subgroup; Normal subgroup). The set of right cosets of a totally ordered group with respect to a convex subgroup is totally ordered by putting if and only if . If is a convex normal subgroup of a totally ordered group, then this order relation turns the quotient group into a totally ordered group.

The system of convex subgroups of a totally ordered group possesses the following properties: a) is totally ordered by inclusion and closed under intersections and unions; b) is infra-invariant, i.e. for any and any one has ; c) if is a jump in , i.e. , , and there is no convex subgroup between them, then is normal in , the quotient group is an Archimedean group and

where is the normalizer of in (cf. Normalizer of a subset); and d) all subgroups of are strongly isolated, i.e. for any finite set in and any subgroup the relation

entails .

An extension of an ordered group by an ordered group (cf. Extension of a group) is an ordered group if the order in is stable under all inner automorphisms of . An extension of an ordered group by a finite group is an ordered group if is torsion-free and if the order in is stable under all inner automorphisms of .

The order type of a countable ordered group has the form , where are the order types of the set of integers and of rational numbers, respectively, and is an arbitrary countable ordinal. Every ordered group is a topological group relative to the interval topology, in which a base of open sets consists of the open intervals

A convex subgroup of an ordered group is open in this topology.

References

[1]	A.I. Kokorin, V.M. Kopytov, "Fully ordered groups" , Israel Program Sci. Transl. (1974) (Translated from Russian)

Comments

If the order relation on the partially ordered group defines a lattice (i.e. for all there exists a greatest lower bound and a least upper bound ), then one speaks of a lattice-ordered group or -group; cf. also Ordered semi-group. These turn up naturally in many branches of mathematics. For a survey of the current state-of-the-art in this field see [a1]–[a3].

References

[a1]	M. Anderson, T. Feil, "Lattice-ordered groups. An introduction" , Reidel (1988)
[a2]	A.M.W. Glass (ed.) W.Ch. Holland (ed.) , Lattice-ordered groups. Advances and techniques , Kluwer (1989)
[a3]	J. Martinez (ed.) , Ordered algebraic structures , Kluwer (1989)