Levi Classes Generated by Nilpotent Groups

Let be a class of all groups G for which the normal closure (x) ^G of every element x belongs to a class . is a Levi class generated by . Let and ₀ be classes of finitely generated nilpotent groups and of torsion-free, finitely generated, nilpotent groups, respectively. We prove that and , and so and . It is shown that quasivarieties and are closed under free products, and that each contains at most one maximal proper subquasivariety. It is also proved that is closed under free products if so is .     

Quasivariety Generated by Free Metabelian and 2-Nilpotent Groups

Let qG be a quasivariety generated by a group G and be a non-Abelian quasivariety of groups with a finite lattice of subquasivarieties. Suppose is contained in a quasivariety generated by the following two groups: a free 2-nilpotent group F₂( ₂) of rank 2 and a free metabelian (i.e., with an Abelian commutant) group F₂( ₂) of rank 2. It is proved that either = qF₂( ₂) or = qF₂( ₂) in this instance.__________Translated from Algebra i Logika, Vol. 44, No. 4, pp. 389–398, July–August, 2005.     

Spin Models and Strongly Hyper-Self-Dual Bose-Mesner Algebras

We introduce the notion of hyper-self-duality for Bose-Mesner algebras as a strengthening of formal self-duality. Let denote a Bose-Mesner algebra on a finite nonempty set X. Fix p X, and let and denote respectively the dual Bose-Mesner algebra and the Terwilliger algebra of with respect to p. By a hyper-duality of , we mean an automorphism of such that for all ; and is a duality of . is said to be hyper-self-dual whenever there exists a hyper-duality of . We say that is strongly hyper-self-dual whenever there exists a hyper-duality of which can be expressed as conjugation by an invertible element of . We show that Bose-Mesner algebras which support a spin model are strongly hyper-self-dual, and we characterize strong hyper-self-duality via the module structure of the associated Terwilliger algebra.     

Finite sublattices in the lattice of clones

The lattice of clones of functions over a k-element set is studied. It is shown that every lattice which is a countable direct product of finite lattices is embedded (up to isomorphism) in and, hence, in for k 4. This directly implies that every finite and any countable residually finite lattice is embedded in , k 4, and that no nontrivial quasi-identity holds in , k 4. A number of particular lattices (which are free in some lattice varieties) embeddable in , k 4,are presented. Translated fromAlgebra i Logika, Vol. 33, No. 5, pp. 514–549, September–October, 1994.     

Rings of regular functions on nilpotent orbits and their covers

Summary LetG be a complex semisimple algebraic group with Lie algebra . Let be a nilpotentG-orbit, its ring of regular functions. We derive a formula for as aG-module and prove some partial results on a cover of . We then relate this formula to various existing multiplicity formulas forK-types in Harish-Chandra bimodules ofG.Supported by National Science Foundation Grant DMS-8505550     

Varieties Defined by Permutations

We continue to study interrelations between permutative varieties and the cyclic varieties defined by cycles of the form . A criterion is given determining whether a cyclic variety is interpretable in . For a permutation without fixed elements, it is stated that a set of primes for which is interpretable in in the lattice is finite. It is also proved that for distinct primes , the Helly number of a type in coincides with dimension of the dual type and equals .     

Lattices of interpretability types of varieties

Let be the set of all primes, the field of all algebraic numbers, and Z the set of square-free natural numbers. We consider partially ordered sets of interpretability types such as , and , where AD is a variety of -divisible Abelian groups with unique taking of the pth root _p(x) for every p , is a variety of -modules over a normal field , contained in , and G_n is a variety of n-groupoids defined by a cyclic permutation (12 ...n). We prove that , and are distributive lattices, with and where ub and ub_f are lattices (w.r.t. inclusion) of all subsets of the set and of finite subsets of , respectively.Deceased.__________Translated from Algebra i Logika, Vol. 44, No. 2, pp. 198–210, March–April, 2005.     

Symmetry of Sections in Fields of Formal Power Series and a Non-Standard Real Line

Let be a field of formal power series with real coefficients, whose supports are well ordered subsets of an Abelian group of cardinality strictly less than . For , we give criteria of a section being symmetric and of a symmetric section being Dedekind. It is proved that an -saturated non-standard real line is isomorphic to some field of the form . For , some consequences are inferred regarding symmetric sections, and the cofinality of banks of the sections.     

Profinite and Continuous Higher K-Theory of Exact Categories, Orders, and Group Rings

Let be a rational prime, an exact category. In this article, we define and study for all , the profinite higher K-theory of , that is as well as , where is the -dimensional mod- Moore space. We study connections between and prove several -completeness results involving these and associated groups including the cases where is the category of finitely generated (resp. finitely generated projective) modules over orders in semi-simple algebras over number fields and p-adic fields. We also define and study continuous K-theory of orders in p-adic semi-simple algebras and show some connection between the profinite and continuous K-theory of .     

Determinants in Triangulated Categories

For a small triangulated category , Bass's K ₁ group is described, and the theorem of the heart is proved. We define the determinant map from to Neeman's , and we compute this map when is the derived category of an Abelian category .     

Soluble Groups and Varieties of <Emphasis Type="Italic">l</Emphasis>-Groups

A sufficient condition is given under which factors of a system of normal convex subgroups of a linearly ordered (l.o.) group are Abelian. Also, a sufficient condition is specified subject to which factors of a system of normal convex subgroups of an l.o. group are contained in a group variety . In particular, for every soluble l.o. group G of solubility index n, n ⩾ 2, factors of a system of normal convex subgroups are soluble l.o. groups of solubility index at most n − 1. It is proved that the variety of all lattice-ordered groups, approximable by linearly ordered groups, does not coincide with a variety generated by all soluble l.o. groups. It is shown that if is any o-approximable variety of l-groups, and if every identity in the group signature is not identically true in , then contains free l.o. groups.Supported by FP “Universities of Russia” grant UR. 04. 01. 001.__________Translated from Algebra i Logika, Vol. 44, No. 3, pp. 355–367, May–June, 2005.     

Operator inequalities and construction of Krein spaces

Let be a Hilbert space. A continuous positive operatorT on uniquely determines a Hilbert space which is continuously imbedded in and for which with the canonical imbedding . A Kreîn space version of this result, however, is not valid in general. This paper provides a necessary and sufficient condition for that a continuous selfadjoint operatorT uniquely determines a Kreîn space ( ) which is continuously imbedded in and for which with the canonical imbedding .     

Minihypers and Linear Codes Meeting the Griesmer Bound: Improvements to Results of Hamada, Helleseth and Maekawa

This article improves results of Hamada, Helleseth and Maekawa on minihypers in projective spaces and linear codes meeting the Griesmer bound.In [10,12],it was shown that any -minihyper, with , where , is the disjoint union of points, lines,..., -dimensional subspaces. For q large, we improve on this result by increasing the upper bound on non-square, to non-square, square, , and (4) for square, p prime, p<3, to . In the case q non-square, the conclusion is the same as written above; the minihyper is the disjoint union of subspaces. When q is square however, the minihyper is either the disjoint union of subspaces, or the disjoint union of subspaces and one subgeometry . For the coding-theoretical problem, our results classify the corresponding codes meeting the Griesmer bound.     

Conditions of Local Asymptotic Normality for Gaussian Stationary Processes

Let {\bold x}[] be a stationary Gaussian process with zero mean and spectral density f, let be the -algebra induced by the random variables {\bold x}[], D(R¹), and let _t, t > 0, be the -algebra induced by the random variables x[],supp [-t,t]. Denote by (f) the Gaussian measure on generated by {\bold x}. Let _t(f) be the restriction of (f) to _t. Let f and g be nonnegative functions such that the measures _t(f) and _t(g) are absolutely continuous. Put

Convex Subgroups of Partially Right-Ordered Groups

We study into the question of whether a partial order can be induced from a partially right-ordered group onto a space of right cosets of w.r.t. some subgroup of . Examples are constructed showing that the condition of being convex for in is insufficient for this. A necessary and sufficient condition (in terms of a subgroup and a positive cone of ) is specified under which an order of can be induced onto . Sufficient conditions are also given. We establish properties of the class of partially right-ordered groups for which is partially ordered for every convex subgroup , and properties of the class of groups such that is partially ordered for every partial right order on and every subgroup that is convex under .     

Equational Theories for Classes of Finite Semigroups

It is proved that there exists an infinite sequence of finitely based semigroup varieties such that, for all i, an equational theory for and for the class of all finite semigroups in is undecidable while an equational theory for and for the class of all finite semigroups in is decidable. An infinite sequence of finitely based semigroup varieties is constructed so that, for all i, an equational theory for and for the class of all finite semigroups in is decidable whicle an equational theory for and for the class of all finite semigroups in is not.