Varieties minimal over representable varieties of lattice-ordered groups

Weinberg showed that the variety of abelian lattice-ordered groups is the minimal nontrivial variety in the lattice of varieties of lattice-ordered groups. Scrimger showed that the abelian variety of lattice-ordered groups has countably infinitely many nonrepresentable covering varieties, and it is now known that his varieties are the only nonrepresentable covers of the abelian variety.

In this paper, a variation of the method used to construct the Scrimger varieties is developed that is shown to produce every nonrepresentable cover of any representable variety. Using this variation, all nonrepresentable covers of any weakly abelian l-variety are specifically identified, as are the nonrepresentable covers of any l-metabelian representable l-variety. In both instances, such il-varieties have only countably infinitely many such covers.

Any nonrepresentable cover of a representable il-variety is shown to be a subvariety of a quasi-representable il-variety as defined by Reilly. The class of these quasi-representable l-varieties is shown to contain the well-known L_n l-varieties and to generalize many of their properties.     

The product of finitely based varieties of lattice-ordered groups

The question as to whether a product of two finitely based varieties of lattice-ordered groups is finitely based is considered. It is proved that varieties and are finitely based; here is a variety of lattice-ordered groups defined by identities [x ⁿ,y ⁿ] =e and [[x,y] z, [x ₁,y ₁] z ₁] =e; is a variety of lattice-ordered nilpotent groups of class s, defined by an identity [x ₁,x ₂,...,x _(s+1)] =e; V is an arbitrary finitely based variety of lattice-ordered groups. Translated fromAlgebra i Logika, Vol. 33, No. 3, pp. 255–263, May–June, 1994.Supported by the Russian Foundation for Fundamental Research, grant No. 93-011-1524.     

On the atomic conditions of lattice-ordered groups

We introduce large convex -subgroups to study the structure of the lattice-ordered groups G whose C(G), P(G) and (G) satisfy atomic conditions, where C(G), P(G) and (G) denote respectively the lattice of all convex -subgroups, the lattice of all polar subgroups and the root system of all regular subgroups of G. In particular, we construct a new torsion class defined as the class of -groups G for which all large prime subgroups are maximal. We prove that the class of hyperarchimedean -groups is properly contained within and that any -group within has the property that any chain of prime subgroups has length at most 2.Received October 7, 2003; accepted in final form June 11, 2004.     

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.     

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.