Solvable covers of the boolean variety of unital ℓ-groups

The varieties of solvable lattice-ordered groups covering the abelian variety were shown independently by Gurchenkov, Reilly, and Darnel to be the Scrimger varieties of ?-groups and the three Medvedev representable covers. In this article, the authors give a parallel characterization of varieties of solvable unital ?-groups which cover the minimal nontrivial variety of boolean unital ?-groups.     

Covers in the lattice of varieties of ℓ-groups

We show that a variety ν in the lattice L of varieties of ℓ-groups has continuum many covers, and that the same is also true of an arbitrary o-approximable variety Χ with the property ν⊆Χ. It is proved that any o-approximable quasivariety Q of ℓ-groups, for which ν⊆Q, has the continuum of covers in the quasivariety lattice Λ. Supported by the RF State Committee of Higher Education. Translated fromAlgebra i Logika, Vol. 37, No. 3, pp. 253–269, May–June, 1998.     

Varieties and Torsion Classes of m-Groups

We prove that every variety of m-groups is a torsion class; find basis of identities for a product variety of m-groups; and show that the product of every finitely based variety of m-groups and a variety of Abelian m-groups is a finitely based variety.     

More covers of the Boolean variety of unital ℓ-groups

Within the lattice of varieties of pseudo MV-algebras, the variety ${\mathcal{B}}$ of Boolean algebras is the least nontrivial variety. Komori identified all varieties of (commutative) MV-algebras that cover ${\mathcal{B}}$ . The authors previously identified all solvable varieties of pseudo MV-algebras that cover ${\mathcal{B}}$ . We will show the existence of continuum many nonsolvable varieties of pseudo MV-algebras that cover ${\mathcal{B}}$ , show that periodically primitive u?-groups cannot generate Boolean covers, and show that all noncommutative varieties that are Boolean covers must be Top Boolean.     

The Greatest Proper Variety of m-Groups

It is proved that the variety of all normal-valued m-groups is the greatest proper subvariety in the lattice of all m-varieties.     

Wreath products of the groups of monotone permutations

We introduce the concept of wreath product of the m-groups of permutations and prove that an m-transitive group of permutations with an m-congruence is embeddable into the wreath product of the suitable m-transitive m-groups of permutations. This implies that an arbitrary m-transitive group in the product of two varieties of m-groups embeds into the wreath product of the suitable m-transitive groups of these varieties.     

Varieties of Birkhoff Systems Part II

This is the second part of a two-part paper on Birkhoff systems. A Birkhoff system is an algebra that has two binary operations ? and + , with each being commutative, associative, and idempotent, and together satisfying x?(x + y) = x+(x?y). The first part of this paper described the lattice of subvarieties of Birkhoff systems. This second part continues the investigation of subvarieties of Birkhoff systems. The 4-element subdirectly irreducible Birkhoff systems are described, and the varieties they generate are placed in the lattice of subvarieties. The poset of varieties generated by finite splitting bichains is described. Finally, a structure theorem is given for one of the five covers of the variety of distributive Birkhoff systems, the only cover that previously had no structure theorem. This structure theorem is used to complete results from the first part of this paper describing the lower part of the lattice of subvarieties of Birkhoff systems.     

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.     

Continuum many top varieties of GMV-algebras and unital ℓ-groups

We prove that there are continuum many top varieties of GMV-algebras and examine the countable lattice of varieties generated by periodic primitive unital ℓ-groups with the unit a rational translation.     

Top Varieties of Generalized MV-Algebras and Unital Lattice-Ordered Groups

In spite of the well-know fact that the system of ?-groups with strong unit (unital ?-groups) does not form a variety, there is a categorical connection between the category of unital ?-groups and the variety of generalized MV-algebras which enables us to naturally export equational machinery and terminology like “variety” from the latter category to the former.

Using this categorical equivalence, we study varieties, or equationally defined classes, and top varieties, varieties above the normal valued variety, of both structures. We generalize Chang's Completeness Theorem for generalized MV-algebras, and formulate some open questions for both structures.     

Abelian groups of monotonic permutations

We study m-transitive representations of Abelian m-groups. Representations are found which mimic a variety A {\mathcal A} of all Abelian m-groups and a variety J {\mathcal J} of m-groups defined by an identity x _*  = x ⁻¹.     

K-theory of twisted Grassmannians

We give a computation of the K-groups of Grassmannians and flag varieties over an arbitrary Noetherian base scheme. We also compute the K-groups of forms of Grassmannians and flag varieties associated to a sheaf of Azumaya algebras. One ingredient in the computation is the extension of the Bott theorem on the cohomology of line bundles on the flag variety (over Q) to a K ₀-Bott theorem valid over arbitrary Noetherian base schemes.Partially supported by the NSF.     

Covers of the Abelian Variety of Generalized MV-Algebras

Using the categorical equivalence of the class of generalized MV-algebras with the class of unital ?-groups, we describe all varieties of symmetric top abelian unital ?-groups that cover the variety  _u? of abelian unital ?-groups. Equivalently, we describe all cover varieties of the variety of MV-algebras, ?, within the variety of generalized MV-algebras admitting only one negation and each of whose maximal ideals is normal. In particular, there are continuum many representable varieties of generalized MV-algebras that cover ?.     

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.