Ro-group
right-ordered group
A group endowed with a total order
such that for all
,
![]() |
If is the positive cone of the
-group
(cf. also
-group), then:
1) ;
2) ;
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.
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) |
Ro-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ro-group&oldid=11213