A paraconsistent extension of Sylvan’s logic

We deal with Sylvan’s logic CC_ω. It is proved that this logic is a conservative extension of positive intuitionistic logic. Moreover, a paraconsistent extension of Sylvan’s logic is constructed, which is also a conservative extension of positive intuitionistic logic and has the property of being decidable. The constructed logic, in which negation is defined via a total accessibility relation, is a natural intuitionistic analog of the modal system S5. For this logic, an axiomatization is given and the completeness theorem is proved. Supported by RFBR grant No. 06-01-00358 and by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-4787.2006.1. __________ Translated from Algebra i Logika, Vol. 46, No. 5, pp. 533–547, September–October, 2007.     

Hochschild cohomologies for associative conformal algebras

We introduce the concept of Hochschild cohomologies for associative conformal algebras. It is shown that the second cohomology group of a conformal Weyl algebra with values in any bimodule is trivial. As a consequence, we derive that the conformal Weyl algebra is segregated in any extension with nilpotent kernel. Supported by RFBR grant No. 05-01-00230 and via SB RAS Integration project No. 1.9. __________ Translated from Algebra i Logika, Vol. 46, No. 6, pp. 688–706, November–December, 2007.     

Modified realizability and predicate logic

Semantics of predicate formulas based on the notion of modified realizability for arithmetic formulas and interpretations of the language of arithmetic in all finite types are considered. For a number of natural constructive interpretations, the corresponding predicate logic of modified realizability is proved to be nonarithmetical. Translated fromMatematicheskie Zametki, Vol. 61, No. 2, pp. 259–269, February, 1997. This research was supported by the Russian Foundation for Basic Research under grant No. 95-01-00416 and by the International Science Foundation under grant No. NFQ000. Translated by V. N. Dubrovsky     

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.     

A weak form of interpolation in equational logic

The notions of a weak interpolation property and of weak amalgamation are introduced. It is proved that in varieties with the congruence extension property, the weak interpolation property is equivalent to the weak amalgamation property. In turn, weak amalgamability of a variety is equivalent to amalgamability of a class of finitely generated simple algebras in this variety. Supported by RFBR (grant Nos. 06-01-00358 and 05-01-04003-NNIOa) and by INTAS (grant No. 04-77-7080). __________ Translated from Algebra i Logika, Vol. 47, No. 1, pp. 94–107, January–February, 2008.     

Generalized computable numerations and nontrivial rogers semilattices

We outline the general approach to the notion of a computable numeration in the frames of which computable numerations, of families of arithmetic sets are studied. Supported by RECO grant NERBCHRXCT 930415 and by RFFR grant No. 096-01-01525 Supported by RECO grant NERBCIPDCT 940615 Translated fromAlgebra i Logika, Vol. 36, No. 6, pp. 621–641, November-December, 1997.     

Spectra of finite linear and unitary groups

The spectrum of a finite group is the set of its element orders. An arithmetic criterion determining whether a given natural number belongs to a spectrum of a given group is furnished for all finite special, projective general, and projective special linear and unitary groups. Supported by RFBR (grant Nos. 08-01-00322 and 06-01-39001) and by the Council for Grants (under RF President) and State Aid of Leading Scientific Schools (project NSh-344.2008.1). __________ Translated from Algebra i Logika, Vol. 47, No. 2, pp. 157–173, March–April, 2008.     

Groups of exponent 60 with prescribed orders of elements

We prove that a group which contains elements of orders 1, 2, 3, 4, 5 and does not contain elements of any other order is locally finite and isomorphic either to an alternating group of degree 6 or to an extension of a nontrivial elementary Abelian 2-group by an alternating group of degree 5. This article was written during my visit to the University of Manitoba, Canada, and supported by RFFR grant No. 99-01-00550. Translated fromAlgebra i Logika, Vol. 39, No. 3, pp. 329–346, May–June, 2000.     

An algebra of skew primitive elements

We study the various term operations on the set of skew primitive elements of Hopf algebras, generated by skew primitive semi-invariants of an Abelian group of grouplike elements. All 1-linear binary operations are described and trilinear and quadrilinear operations are given a detailed treatment. Necessary and sufficient conditions for the existence of multilinear operations are specified in terms of the property of particular polynomials being linearly dependent and of one arithmetic condition. We dub the conjecture that this condition implies, in fact, the linear dependence of the polynomials in question and so is itself sufficient. Supported by RFFR grant No. 95-01-01356, and by the National Society of researchers, Mexico (exp. 18740, 1997). Translated fromAlgebra i Logika, Vol. 37, No. 2, pp. 181–223, March–April, 1998.     

On a width of verbal subgroups of certain free constructions

We study into widths of verbal subgroups of HNN-extensions, and of groups with one defining relation. It is proved that if a group G^* is an HNN-extension and the connected subgroups in G^* are distinct from a base of the extension, then every verbal subgroup V(G^*) has infinite width relative to a finite proper set V of words. A similar statement is proven to hold for groups presented by one defining relation and ≥3 generators. to Yurii I. Merzlyakov dedicated Supported by RFFR grant No. 93-01-01513. Translated fromAlgebra i Logika, Vol. 36, No. 5, pp. 494–517, September–October, 1997.     

The Hilbert pairing for formal groups over σ-rings

In the paper, formal groups over the rings of integers of σ-fields are studied. These fields were constructed by the first author in a previous paper. They are a generalization of the inertia field of a classical local field to an arbitrary complete discrete valuation field of characteristic zero. An analog of Honda’s theory for such formal groups is constructed. The arithmetic of the group of points in an extension of a σ-field that contains sufficiently many torsion points is studied. Using the classification of formal groups and the arithmetic results obtained, an explicit formula for the Hilbert pairing for formal groups over σ-fields is proved. Bibliography: 16 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 319, 2004, pp. 5–58.     

An Adjacency Criterion for the Prime Graph of a Finite Simple Group

For every finite non-Abelian simple group, we give an exhaustive arithmetic criterion for adjacency of vertices in a prime graph of the group. For the prime graph of every finite simple group, this criterion is used to determine an independent set with a maximal number of vertices and an independent set with a maximal number of vertices containing 2, and to define orders on these sets; the information obtained is collected in tables. We consider several applications of these results to various problems in finite group theory, in particular, to the recognition-by-spectra problem for finite groups. Supported by RFBR grant No. 05-01-00797; by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-2069.2003.1; by the RF Ministry of Education Developmental Program for Scientific Potential of the Higher School of Learning, project No. 8294; by FP “Universities of Russia,” grant No. UR.04.01.202; and by Presidium SB RAS grant No. 86-197. __________ Translated from Algebra i Logika, Vol. 44, No. 6, pp. 682–725, November–December, 2005.     

Siegel-Shidlovsky method in <Emphasis Type="Italic">p</Emphasis>-adic domain

This paper presents a review of the arithmetic properties of F-series. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 6, pp. 221–230, 2005.     

Stable valued fields

We are concerned with a class of valued fields, called stable. We propound an extension of a notion in the monograph by S. Bosch, U. Güntzer, and R. Remmert (Non-Archimedean Analysis. A Systematic Approach to Rigid Analytic Geometry, Springer, Berlin (1984)), namely, that of a (ultrametric) norm on groups, rings, algebras, and vector spaces, to the case where the value of the norm is taken from an arbitrary (not necessarily Archimedean) linearly ordered Abelian group (using — as in the general theory of valuations — the version of a logarithmic norm). Our main result extends Proposition 6 in the cited monograph to the general case, thereby making it possible to use the technique of Cartesian spaces to deliver further results on stable valued fields. Supported by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools via project NSh-4787.2006.1 and by RFBR grant No. 05-01-00481. __________ Translated from Algebra i Logika, Vol. 46, No. 6, pp. 707–728, November–December, 2007.     

Some estimates in theL p metric in the central limit theorem for additive functions

The convergence of the value distributions of a normalized sequence of strongly additive arithmetic functions to the standard normal law in theL _p metric is considered. An asymptotic formula for the variance is obtained. In both cases, the remainders are expressed in terms of the third absolute moment of the additive function. Vilnius University, Naugarduko 24, 2006 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 39, No. 1, pp. 74–80, January–March, 1999. Translated by A. Mačiulis     

A Note on Linear Extension of Into-Isometries Between Two Unit Spheres of Atomic AL^p-Space （0 〈 p 〈∞）

In this paper, we shall present a short and simple proof on the isometric linear extension problem of into-isometries between two unit spheres of atomic abstract L^p-spaces （0 〈 p 〈 ∞）.     

Hahn–Banach extension theorems for multifunctions revisited

Several generalizations of the Hahn–Banach extension theorem to K-convex multifunctions were stated recently in the literature. In this note we provide an easy direct proof for the multifunction version of the Hahn–Banach–Kantorovich theorem and show that in a quite general situation it can be obtained from existing results. Then we derive the Yang extension theorem using a similar proof as well as a stronger version of it using a classical separation theorem. Moreover, we give counterexamples to several extension theorems stated in the literature. Dedicated to Jean-Paul Penot with the occasion of his retirement.     

Definable predicates of standardness in internal set theory

It is shown that Nelson’s internal set theory IST has no definable predicate that is a proper extension of the standardness predicate and satisfies the carry-over, idealization, and standardization principles. Translated fromMatematicheskie Zametki, Vol. 66, No. 6, pp. 803–809, December, 1999.     

Σ-Definability of countable structures over real numbers, complex numbers, and quaternions

We study Σ-definability of countable models over hereditarily finite {ie193-01} superstructures over the field ℝ of reals, the field ℂ of complex numbers, and over the skew field ℍ of quaternions. In particular, it is shown that each at most countable structure of a finite signature, which is Σ-definable over {ie193-02} with at most countable equivalence classes and without parameters, has a computable isomorphic copy. Moreover, if we lift the requirement on the cardinalities of the classes in a definition then such a model can have an arbitrary hyperarithmetical complexity, but it will be hyperarithmetical in any case. Also it is proved that any countable structure Σ-definable over {ie193-03}, possibly with parameters, has a computable isomorphic copy and that being Σ-definable over {ie193-04} is equivalent to being Σ-definable over {ie193-05}. Supported by RFBR-DFG, grant No. 06-01-04002 DFGa. __________ Translated from Algebra i Logika, Vol. 47, No. 3, pp. 335–363, May–June, 2008.