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
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.
|||A.I. Kokorin, V.M. Kopytov, "Fully ordered groups" , Israel Program Sci. Transl. (1974) (Translated from Russian)|
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].
|[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)|
Ordered group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ordered_group&oldid=16970