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Right-ordered group

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A group on whose set of elements a total order (cf. Totally ordered group) is defined such that for all the inequality implies . The set of positive elements of is a pure (i.e. ) linear (i.e. ) sub-semi-group. Every pure linear sub-semi-group of an arbitrary group defines a right order, namely if and only if .

The group of automorphisms of a totally-ordered set can be right ordered in a natural manner. Every right-ordered group is order-isomorphic to some subgroup of for a suitable totally-ordered set (cf. [1]). An Archimedean right-ordered group, i.e. a right-ordered group for which Archimedes' axiom holds (cf. Archimedean group), is order-isomorphic to a subgroup of the additive group of real numbers. In contrast with (two-sided) ordered groups, there are non-commutative right-ordered groups without proper convex subgroups (cf. Convex subgroup). The class of right-ordered groups is closed under lexicographic extension. The system of all convex subgroups of a right-ordered group is totally ordered with respect to inclusion and is complete. This system is solvable (cf. also Solvable group) if and only if for any positive elements there is a natural number such that . If the group has a solvable subgroup system whose factors are torsion-free, then can be right-ordered in such a way that all subgroups in become convex. In a locally nilpotent right-ordered group the system of convex subgroups is solvable.

A group can be right-ordered if and only if for any finite system

of elements of there are numbers , , such that the semi-group generated by the set does not contain the identity element of .

Every lattice ordering of a group is the intersection of some of its right-orderings (cf. Lattice-ordered group).

References

[1] A.I. Kokorin, V.M. Kopytov, "Fully ordered groups" , Israel Program Sci. Transl. (1974) (Translated from Russian)
[2] R.B. Mura, A. Rhemtulla, "Orderable groups" , M. Dekker (1977)


Comments

A group that admits a total order such that with this order becomes a right-ordered group, is called right-orderable. Such an order on is called a right order or right ordering.

Some more concepts and results concerning right-ordered groups can be found in [a1][a4].

References

[a1] M. Anderson, T. Feil, "Lattice-ordered groups. An introduction" , Reidel (1988) pp. 35; 38ff
[a2] A.M.W. Glass (ed.) W.Ch. Holland (ed.) , Lattice-ordered groups. Advances and techniques , Kluwer (1989)
[a3] W.B. Powell, "Universal aspects of the theory of lattice-ordered groups" J. Martinez (ed.) , Ordered Algebraic Structures , Kluwer (1989) pp. 11–50
[a4] M.R. Darnell, "Recent results on the free lattice ordered group over a right-orderable group" J. Martinez (ed.) , Ordered Algebraic Structures , Kluwer (1989) pp. 51–57
How to Cite This Entry:
Right-ordered group. V.M. Kopytov (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Right-ordered_group&oldid=15087
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098