Ordered group

A group $ G $ with an order relation $ \leq $ such that for any $ a , b , x , y $ in $ G $ the inequality $ a \leq b $ entails $ x a y \leq x b y $. 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 $ G $ into an ordered group $ H $ is a homomorphism $ \phi $ of $ G $ into $ H $ such that $ x \leq y $, $ x , y \in G $, implies $ x \phi \leq y \phi $ in $ H $. 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 $ G $ with respect to a convex subgroup $ H $ is totally ordered by putting $ H x \leq H y $ if and only if $ x \leq y $. If $ H $ is a convex normal subgroup of a totally ordered group, then this order relation turns the quotient group $ G / H $ into a totally ordered group.

The system $ \Sigma (G) $ of convex subgroups of a totally ordered group possesses the following properties: a) $ \Sigma (G) $ is totally ordered by inclusion and closed under intersections and unions; b) $ \Sigma (G) $ is infra-invariant, i.e. for any $ H \in \Sigma (G) $ and any $ x \in G $ one has $ x ^ {-1} H x \in \Sigma (G) $; c) if $ A < B $ is a jump in $ \Sigma (G) $, i.e. $ A , B \in \Sigma (G) $, $ A \subset B $, and there is no convex subgroup between them, then $ A $ is normal in $ B $, the quotient group $ B / A $ is an Archimedean group and

$$ [ [ N _ {G} (B) , N _ {G} (B) ] , B ] \subseteq A , $$

where $ N _ {G} (B) $ is the normalizer of $ B $ in $ G $( cf. Normalizer of a subset); and d) all subgroups of $ \Sigma (G) $ are strongly isolated, i.e. for any finite set $ x, g _ {1} \dots g _ {n} $ in $ G $ and any subgroup $ H \in \Sigma (G) $ the relation

$$ x \cdot g _ {1} ^ {-1} x g _ {1} \dots g _ {n} ^ {-1} x g _ {n} \in H $$

entails $ x \in H $.

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

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

$$ ( a , b ) = \{ {x \in G } : {a < x < b } \} . $$

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 $ a,b \in G $ there exists a greatest lower bound $ a \wedge b $ and a least upper bound $ a \lor b $), then one speaks of a lattice-ordered group or $ l $- 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)