A group endowed with a total order such that for all ,
If is the positive cone of the -group (cf. also -group), then:
3) . If, in a group , there is a subset satisfying 1)–3), then can given the structure of a -group with positive cone by a setting if and only if . The positive cone of a -group is isolated, i.e., .
The group of order automorphisms of a totally ordered set can be turned into a -group by defining the following relation on it. Let be any well ordering on : . Let and let be the first (with respect to ) element in . Then is a -group with respect to the order with positive cone
Any -group is isomorphic to a subgroup of the -group for some totally ordered set . There exist simple -groups whose finitely generated subgroups coincide with the commutator subgroup. The class of all groups that can be turned into a -group is a quasi-variety, i.e., it is defined by a system of formulas of the form:
where , 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 of convex subgroups of a -group is a complete chain. It can be non-solvable, non-infra-invariant and non-normal. There exist non-Abelian -groups without proper convex subgroups.
A -group is Archimedean if for any positive elements there exists a positive integer such that . An Archimedean -group is order-isomorphic to some subgroup of the additive group of real numbers with the natural order. The class of Conradian -groups, i.e., -groups for which the system is subnormal and the quotient groups of the jumps of are Archimedean, is well investigated.
|[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)|
Ro-group. V.M. Kopytov (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Ro-group&oldid=11213