Generalized commutativity of lattice-ordered groups II

Commutative \({\ell}\)-groups G (in which for all \({x, y \in G, xy = yx}\)) were studied long ago. This was then generalized to the study of \({\ell}\)-groups G in which for a given integer n and for all \({x, y \in G, x^{n}y^{n} = y^{n}x^{n}}\). It was then discovered that if for all \({x, y \in G}\), both \({x^{n}y^{n} = y^{n}x^{n}}\) and \({x^{m}y^{m} = y^{m}x^{m}}\) for two different integers m, n, then also \({x^{d}y^{d} = y^{d}x^{d}}\), where d is the greatest common divisor of m, n.     

Nilpotent lattice-ordered groups

Translated from Matematicheskie Zametki, Vol. 45, No. 1, pp. 72–79, January, 1989.     

Engel lattice-ordered groups

An Engel l-group generating a proper normal-valued l-variety is shown to be o-approximable. It is also established that for every proper normal-valued l-varietyF, the class (F) of Engel l-groups from F is a torsion class.Translated fromAlgebra i Logika, Vol. 34, No. 4, pp. 398–404, July-August, 1995.     

Ordering groups and validity in lattice-ordered groups

An inductive characterization is given of the subsets of a group that extend to the positive cone of a right order on the group. This characterization is used to relate validity of equations in lattice-ordered groups (?-groups) to subsets of free groups that extend to the positive cone of a right order. As a consequence, new proofs are obtained of the decidability of the word problem for free ?-groups and generation of the variety of ?-groups by the ?-group of automorphisms of the real line. An inductive characterization is also given of the subsets of a group that extend to the positive cone of an order on the group. In this case, the characterization is used to relate validity of equations in varieties of representable ?-groups to subsets of relatively free groups that extend to the positive cone of an order.     

Signatures andS-discrete lattice-ordered groups

The central theme of this article is the approximation of lattice-ordered groups (l-groups) first by Specker groups and, subsequently, by the so-calledS-discretel-groups. The sense of these approximations is made precise via the notion of a signature, defined below, and by that ofa ^*-subgroups. Sample result: ifG is a projectablel-group then it has anl-subgroupH which is Specker and for which the mapPPH defines a boolean isomorphism between the algebras of polars ofG andH.Presented by L. Fuchs.This article was written while this author was a Stouffer Visiting Professor at the University of Kansas. He wishes to thank the members of the Mathematics Department of that institution for their hospitality.     

Weakly abelian lattice-ordered groups

Every nilpotent lattice-ordered group is weakly Abelian; i.e., satisfies the identity . In 1984, V. M. Kopytov asked if every weakly Abelian lattice-ordered group belongs to the variety generated by all nilpotent lattice-ordered groups [The Black Swamp Problem Book, Question 40]. In the past 15 years, all attempts have centred on finding counterexamples. We show that two constructions of weakly Abelian lattice-ordered groups fail to be counterexamples. They include all preiously considered potential counterexamples and also many weakly Abelian ordered free groups on finitely many generators. If every weakly Abelian ordered free group on finitely many generators belongs to the variety generated by all nilpotent lattice-ordered groups, then every weakly Abelian lattice-ordered group belongs to this variety. This paper therefore redresses the balance and suggests that Kopytov's problem is even more intriguing.

     

Extensions of lattice-ordered groups

A number of conditions are specified which are sufficient for totally ordered groups with polycyclic factor group to contain a finite normal series of convex subgroups whose factors possess good enough properties. In any case studying such totally ordered groups is reduced to treating extensions of these groups as well as their virtually o-equivalent extensions. The concept of a virtually o-equivalent extension is a particular case of the notion of an Archimedean extension. Supported by RFBR project No. 03-01-00320. Translated from Algebra i Logika, Vol. 47, No. 5, pp. 529–540, September–October, 2008.     

Right-ordered and free lattice-ordered groups

The free -groups over nilpotent groups are considered. The upper bound of -solvable length is found for an -group over a nilpotent group of finite rank; the lower bound of solvable length is given for a free -group over a finitely generated nilpotent group.Translated fromAlgebra i Logika, Vol. 34, No. 1, pp. 33–40, January–February, 1995.Supported by the Russian Foundation for Fundamental Research, grant No. 93-011-1524.     

Torsion classes of Specker lattice ordered groups

In this paper we investigate the relations between torsion classes of Specker lattice ordered groups and torsion classes of generalized Boolean algebras.