Ro-group

right-ordered group

A group $ G $ endowed with a total order $ \cle $ such that for all $ x,y,z \in G $,

$$ x \cle y \Rightarrow xz \cle yz. $$

If $ P = P ( G ) = \{ {x \in G } : {x \cge e } \} $ is the positive cone of the $ ro $- group $ G $( cf. also $ l $- group), then:

1) $ P \cdot P \subseteq P $;

2) $ P \cap P ^ {- 1 } = \{ e \} $;

3) $ P \cup P ^ {- 1 } = G $. If, in a group $ G $, there is a subset $ P $ satisfying 1)–3), then $ G $ can given the structure of a $ ro $- group with positive cone $ P $ by a setting $ x \cle y $ if and only if $ yx ^ {- 1 } \in P $. The positive cone of a $ ro $- group is isolated, i.e., $ x ^ {n} \in P \Rightarrow x \in P $.

The group of order automorphisms $ { \mathop{\rm Aut} } ( X ) $ of a totally ordered set $ \{ X; \cle \} $ can be turned into a $ ro $- group by defining the following relation $ \cle $ on it. Let $ \prec $ be any well ordering on $ X $: $ x _ {1} \prec \dots \prec x _ \alpha \prec \dots $. Let $ \varphi \in { \mathop{\rm Aut} } ( X ) $ and let $ x _ \alpha $ be the first (with respect to $ \prec $) element in $ \{ {x \in X } : {x \varphi \neq x } \} $. Then $ A ( X ) $ is a $ ro $- group with respect to the order with positive cone

$$ P \subset A ( X ) = \left \{ {\varphi \in { \mathop{\rm Aut} } ( X ) } : {x _ \alpha \varphi \cge x _ \alpha } \right \} . $$

Any $ ro $- group is isomorphic to a subgroup of the $ ro $- group $ { \mathop{\rm Aut} } ( X ) $ for some totally ordered set $ X $. There exist simple $ ro $- groups whose finitely generated subgroups coincide with the commutator subgroup. The class of all groups that can be turned into a $ ro $- group is a quasi-variety, i.e., it is defined by a system of formulas of the form:

$$ \forall x _ {1} \dots x _ {n} : $$

$$ ( w _ {1} ( x _ {1} \dots x _ {n} ) = e \& \dots \& w _ {m} ( x _ {1} \dots x _ {n} ) = e ) \Rightarrow $$

$$ \Rightarrow w ( x _ {1} \dots x _ {n} ) = e, $$

where $ w $, $ w _ {i} $ are the group-theoretical words. This class is closed under formation of subgroups, Cartesian and free products, and extension, and is locally closed.

The system $ {\mathcal C} ( G ) $ of convex subgroups of a $ ro $- group $ G $ is a complete chain. It can be non-solvable, non-infra-invariant and non-normal. There exist non-Abelian $ ro $- groups without proper convex subgroups.

A $ ro $- group $ G $ is Archimedean if for any positive elements $ x,y \in G $ there exists a positive integer $ n $ such that $ x ^ {n} > y $. An Archimedean $ ro $- group is order-isomorphic to some subgroup of the additive group $ \mathbf R $ of real numbers with the natural order. The class of Conradian $ ro $- groups, i.e., $ ro $- groups for which the system $ {\mathcal C} ( G ) $ is subnormal and the quotient groups of the jumps of $ {\mathcal C} ( G ) $ are Archimedean, is well investigated.

References

[a1]	V.M. Kopytov, N.Ya. Medvedev, "The theory of lattice-ordered groups" , Kluwer Acad. Publ. (1994) (In Russian)
[a2]	R.T.B. Mura, A.H. Rhemtulla, "Orderable groups" , M. Dekker (1977)