Primitive connected theories

We prove the quantifier-elimination theorem for so-called primitive connected theories, exemplified by theories of modules. The theorem generalizes the well-known Baur-Monk-Garavaglia theorem on the elimination of quantifiers in the model theory of modules. The definition of a class of primitive connected theories, as distinct from modules. is not supposed to impose any conditions on a type of axioms that would specify those theories. Dedicated to the 60th birthday of Academician Yu. L. Ershov Supported by RFFR grant No. 99-01-00600. Translated fromAlgebra i Logika, Vol. 39, No. 2, pp. 145–169, March–April, 2000.     

Lower central series of sylow 2-subgroups of symplectic and orthogonal groups over ℤ2 m

Lower central series are constructed for 2-subgroups of orthogonal and symplectic groups over the ring of residue classes of integers modulo 2^m, treated as exceptional in the previous accounts. Nilpotency classes are found out. Supported by RFFR grant No. 96-01-00409, and by the Krasnoyarsk Foundation for Sciences, grant No. 5f0107. Translated fromAlgebra i Logika, Vol. 37, No. 4, pp. 413–431, July–August, 1998.     

Generation of finite groups by the class of conjugate Abelian subgroups

Using the basic theorem on the classification of finite simple groups, we answer one of the questions concerning the generation of finite groups by the class of conjugate Abelian subgroups. Supported by RFFR grant No. 93-01-01529. Translated fromAlgebra i Logika, Vol. 35, No. 3, pp. 288–293, May–June, 1996.     

Coedge regular graphs without 3-stars

We describe coedge regular graphs such that antineighborhoods of their vertices are coedge regular graphs with the same value of the parameterμ. As a consequence of the main theorem, we obtain a classification of coedge regular graphs without 3-stars. Translated fromMatematicheskie Zametki, Vol. 60, No. 4, pp. 495–503, October, 1996. This research was supported by the Russion Foundation for Basic Research under grants No. 93-01-01529 and No. 94-01-00802a.     

Isomorphically rigid algebras

Various classes of non-associative algebras possessing the property of being rigid under abstract isomorphisms are studied. Supported by RFBR grant No. 06-01-00159a. __________ Translated from Algebra i Logika, Vol. 46, No. 4, pp. 483–502, July–August, 2007.     

Conditional identity calculus and conditioned rational equivalence

We construct a conditional identity calculus (similar to the Birkhoff identity calculus), which complies with the concept of truth for a conditional identity on a universal algebra. The relationship is studied between the isomorphism of embedding categories of conditional varieties and the conditioned rational equivalence of these varieties. As applications, we describe invariants for the relations ‘is conditional rational equivalent’ and ‘is similar’ on finite universal algebras. Supported by RFFR grant No. 93-01-01520. Translated fromAlgebra i Logika, Vol. 37, No. 4, pp. 432–459, July–August, 1998.     

Partially commutative metabelian groups: centralizers and elementary equivalence

For partially commutative metabelian groups, annihilators of elements of commutator subgroups are described; canonical representations of elements are defined; approximability by torsion-free nilpotent groups is proved; centralizers of elements are described. Also, it is proved that two partially commutative metabelian groups have equal elementary theories iff their defining graphs are isomorphic, and that every partially commutative metabelian group is embeddable in a metabelian group with decidable universal theory. Dedicated to V. N. Remeslennikov on the occasion of his 70th birthday Supported by RFBR (project No. 09-01-00099). Translated from Algebra i Logika, Vol. 48, No. 3, pp. 309–341, May–June, 2009.     

Stably Definable Classes of Theories

A question is studied as to which properties (classes) of elementary theories can be defined via generalized stability. We present a topological account of such classes. It is stated that some well-known classes of theories, such as strongly minimal, o-minimal, simple, etc., are stably definable, whereas, for instance, countably categorical, almost strongly minimal, ω-stable ones, are not. __________ Translated from Algebra i Logika, Vol. 44, No. 5, pp. 583–600, September–October, 2005. Supported by RFBR grant Nos. 02-01-00540 and 05-01-00411, and by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-2069.2003.1.     

A property of creative sets

A new approach to the study of creative sets using the notion of a table is offered. Making use of tables conforming to recursively enumerable sets, novel properties of creative sets are established. Harrington's theorem on the definability of creative sets in the lattice of recursively enumerable sets is proved, and we reprove Lachlan's theorem which states that one of the factors in a direct product of creative sets is again creative. Supported by RFFR grant No. 93-01-16014. Translated fromAlgebra i Logika, Vol. 35, No. 3, pp. 294–307, May–June, 1996.     

Classification up to cobordism of manifolds with simple action of ℤ/p

We deal with quasi-complex manifolds with an action of the group ℤ/p such that the set of fixed points of this action has a trivial normal bundle. The set of cobordism classes of these manifolds is described in terms of the coefficients of the formal group of geometric cobordisms and in terms of characteristic numbers. We also establish the relationship between this work and relevant papers containing a solution of this problem in some particular cases. Translated fromMatematicheskie Zametki, Vol. 63, No. 2, pp. 260–268, February, 1998. The author is greatly indebted to his advisor Professor V. M. Bukhshtaber for his active interest in this work and useful discussions. This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-01404.     

Lattices with unique irreducible decompositions

We define the concept of a minimal decomposition in a lattice, and prove that all the currently known lattices with unique irreducible decompositions are in fact lattices with minimal ones. Also, the characterization of a class of lattices with minimal decompositions is given. A new proof of the Crawley-Dilworth characterization theorem for the class of coalgebraic strongly coatomic lattices with unique irreducible decompositions obtains as a consequence. In memory of Victor A. Gorbunov Supported by the RF State Committee of Higher Education (1998 project), by RFFR grant No. 99-01-00485, and jointly by RFFR and DFG grants Nos. 96-01-00097 and 436113/2670 respectively. Translated fromAlgebra i Logika, Vol. 39, No. 1, pp. 93–103, January–February, 2000.     

Integral characteristics in the inverse problem of magnetotelluric sounding

This article examines some aspects of the one-dimensional inverse problem of magnetotelluric sounding. A uniqueness theorem is proved in the presence ofS-surfaces. A numerical algorithm based on transformation formulas is proposed. This research was partially supported by Russian Foundation for Basic Research (grant No. 96-01-00410) and by the State Scientific-Technical Program “Future Information Technologies” (grant No. 0201.06.010). Translated from Chislennye Metody v Matematicheskoi Fizike, Moscow State University, pp. 53–66, 1998.     

Horn classes of predicate systems and varieties of partial algebras

We propound an approach through which techniques of the theory of quasivarieties of predicate systems are brought to bear on partial algebras. For every partial algebra A, two predicate representations are treated. The first is the graph of A whose basic operations are graphs of the basic operations on A. The second representation results from the graph of A by adding domains of the operations on A to its basic relations. Studying partial algebras from various perspectives makes it necessary to deal with different equality semantics. Here we present a general definition of semantics that stretches over such instances as weak semantics, Evans’ semantics. Kleene semantics, and strong semantics. On a set of all semantics, the preorder is induced in increasing “force,” and it is proved that certain of the properties of varieties of partial algebras in a given semantics are individuated by the position it takes in that set. We argue that every variety of partial algebras, in any semantics, is in correspondence with a Horn class of predicate systems which admits a generation operator and is closed under direct limits and retracts. For such classes we prove analogs of the Birkhoff theorem on subdirect decompositions and of the Taylor theorem on residual smallness. Therefore, these are also applicable to varieties of partial algebras in arbitrary semantics. Supported through the RF State Committee of Higher Education (1998 project), jointly by RFFR and DFG grants Nos. 96-01-00097 and 436113/2670, and also through FP “Integration” project No. 274. Translated fromAlgebra i Logika, Vol. 39, No. 1, pp. 23–46, January–February, 2000.     

On maxima of partial samples in Gaussian sequences with pseudo-stationary trends

We consider a Gaussian stationary sequence added by a pseudo-stationary trend and prove a limit theorem for joint distribution of its maximum and maximum of its subsequence. Supported by RFFI grants 07-01-00077 and 06-01-00454. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 1, pp. 58–67, January–March, 2007.     

Groups of formal power series are fully orderable

We construct an example of a fully orderable group that is not locally solvable. It is also shown that a free group is embedded in a fully orderable group. To meet these ends, use is made of a group of invertible formal power series with zero free term under composition. Supported by RFFR grant No. 96-01-00088. Translated fromAlgebra i Logika, Vol. 37, No. 3, pp. 301–319, May–June, 1998.     

Complete theories with finitely many countable models. II

Previously, we obtained a syntactic characterization for the class of complete theories with finitely many pairwise non-isomorphic countable models [1]. The most essential part of that characterization extends to Ehrenfeucht theories (i.e., those having finitely many (but more than 1) pairwise non-isomorphic countable models). As the basic parameters defining a finite number of countable models, Rudin-Keisler quasiorders are treated as well as distribution functions defining the number of limit models for equivalence classes w.r.t. these quasiorders. Here, we argue to state that all possible parameters given in the characterization theorem in [1] are realizable. Also, we describe Rudin-Keisler quasiorders in arbitrary small theories. The construction of models of Ehrenfeucht theories with which we come up in the paper is based on using powerful digraphs which, along with powerful types in Ehrenfeucht theories, always locally exist in saturated models of these theories. Supported by RFBR grant Nos. 02-01-00258 and 05-01-00411. __________ Translated from Algebra i Logika, Vol. 45, No. 3, pp. 314–353, May–June, 2006.     

Some properties of autostable models

Established are (1) a nonuniform criterion for the stability of models in terms of enumeration reducibility of constructivizations; (2) a criterion for the autostability of certain particular classes of models close to algebraic number fields; (3) a uniform autostability of each 1-constructive model that is autostable. Supported by RFFR grant No. 096-01-01525 and by ISF grant NQ 6000. Translated fromAlgebra i Logika, Vol. 35, No. 6, pp. 685–698, November–December, 1996.     

Strictly pseudoconvex domains and algebraic varieties

In the paper, we consider applications of strictly pseudoconvex domains to the problems of algebraicity and rationality. We give a new proof of the Kodaira theorem on the algebraicity of a surface and we also prove a multidimensional version of this theorem. Theorems analogous to the Hodge index theorem and the Lefschetz theorem about (1, 1)-classes are obtained for strictly pseudoconvex domains. Conjectures on the geometry of strictly pseudoconvex domains on algebraic surfaces are formulated. Translated fromMatematicheskie Zametki, Vol. 60, No. 3, pp. 414–422, September, 1996. This research was supported by the Russian Foundation for Basic Research under grant No. 93-01-00225 and by the International Science Foundation under grant No. 508.