Constructivizability criterion for an Abelian group

We establish a criterion determining whether an Abelian group is constructivizable, which is then used to prove that for any natural number r≥1, there exists a principal computable enumeration λ of the class K_r of all constructive groups whose torsion-free ranks are distinct from zero and do not exceed r. Translated fromAlgebra i Logika, Vol. 38, No. 6, pp. 743–760, November–December, 1999.     

Fields with continuous local elementary properties. III

In the first part of this work (see [1]), fields of characteristic 0 were treated. Here we prove analogs of the theorems from that article for the case of perfect RC^*-fields 〈F, W〉 of characteristic p>0 with a Boolean family W of valuation rings R of a field F, such that Henselizations are tame. Supported by RFFR grant No. 96-01-01552. Translated fromAlgebra i Logika, Vol. 36, No. 6, pp. 642–656, November–December, 1997.     

Order topology in vector lattices with closure by one step

An order topology in vector lattices and Boolean algebras is studied under the additional condition of “closure by one step” that generalizes the well-known “regularity” property of Boolean algebras and K-spaces. It is proved that in a vector lattice or a Boolean algebra possessing such a property there exists a basis of solid neighborhoods of zero with respect to an order topology. An example of a Boolean algebra without basis of solid neighborhoods of zero (an algebra of regular open subsets of the interval (0, 1)) is given. Bibliography: 3 titles. Translated fromProblemy Matematicheskogo Analiza, No. 15 1995, pp. 213–220.     

Complexity of Boolean algebras and their Scott rank

We deal with problems associated with Scott ranks of Boolean algebras. The Scott rank can be treated as some measure of complexity of an algebraic system. Our aim is to propound and justify the procedure which, given any countable Boolean algebra, will allow us to construct a Boolean algebra of a small Scott rank that has the same natural algebraic complexity as has the initial algebra. In particular, we show that the Scott rank does not always serve as a good measure of complexity for the class of Boolean algebras. We also study into the question as to whether or not a Boolean algebra of a big Scott rank can be decomposed into direct summands with intermediate ranks. Examples are furnished in which Boolean algebras have an arbitrarily big Scott rank such that direct summands in them either have a same rank or a fixed small one, and summands of intermediate ranks are altogether missing. This series of examples indicates, in particular, that there may be no nontrivial mutual evaluations for the Scott and Frechet ranks on a class of countable Boolean algebras. Supported by RFFR grant No. 99-01-00485, by a grant for Young Scientists from SO RAN, 1997, and by the Federal Research Program (FRP) “Integration”. Translated fromAlgebra i Logika, Vol. 38, No. 6, pp. 643–666, November–December, 1999.     

On the Ershov Upper Semilattice

We find some links between -reducibility and T-reducibility. We prove that (1) if a quasirigid model is strongly -definable in a hereditarily finite admissible set over a locally constructivizable B-system, then it is constructivizable; (2) every abelian p-group and every Ershov algebra is locally constructivizable; (3) if an antisymmetric connected model is -definable in a hereditarily finite admissible set over a countable Ershov algebra then it is constructivizable.     

Condition for inducing topology in a vector lattice by the order completion

Properties of an order topology in vector lattices and Boolean algebras are studied. The main result is the following: in a vector lattice or a Boolean algebra with the condition of “closure by one step” (a generalization of the well-known “regularity” property of Boolean algebras and K-spaces) the order topology is induced by the topology of its Dedekind completion. Bibliography: 4 titles. Translated fromProblemy Matematicheskogo Analiza, No. 16, 1997, pp. 204–207.     

Some infinite groups with strongly embedded subgroup

An involution i of a group G is said to be finite if |ii^g|<∞ for all g ∃ G. Suppose that G contains a finite involution and an infinite elementary Abelian 2-subgroup S and, moreover, the normalizer H=N_G(S)=SλT is strongly embedded in G and is a Frobenius group with locally cyclic complement T. It is proved that G is isomorphic to L₂(Q) over a locally finite field Q of characteristic 2. In particular, part (a) of Question 10.76 raised by Shunkkov in the Kourovka Notebook is answered in the affirmative. Supported by RFFR grant No. 99-01-00542. Translated fromAlgebra i Logika, Vol. 39, No. 5, pp. 602–617, September–October, 2000.     

Non-self-equivalent constructivization of atomic Boolean algebras

In the paper we prove that any constructivizable infinite atomic Boolean algebra has ₀ non-self-equivalent constructivizations.Translated from Matematicheskie Zametki, Vol. 19, No. 6, pp. 853–858, June, 1976.     

δ-Derivations of simple finite-dimensional Jordan superalgebras

We describe non-trivial δ-derivations of semisimple finite-dimensional Jordan algebras over an algebraically closed field of characteristic not 2, and of simple finite-dimensional Jordan superalgebras over an algebraically closed field of characteristic 0. For these classes of algebras and superalgebras, non-zero δ-derivations are shown to be missing for δ ≠ 0, 1/2, 1, and we give a complete account of 1/2-derivations. Supported by RFBR grant No. 05-01-00230 and by RF Ministry of Education and Science grant No. 11617. __________ Translated from Algebra i Logika, Vol. 46, No. 5, pp. 585–605, September–October, 2007.     

Nil-elements of index 2 in Mal’tsev algebras

It is proved that a Mal'tsev algebra over an associative commutative ring with 1, which contains 1/6 and is generated by a finite tuple of nil-elements of index 2, is nilpotent, and that an ideal of the Mal'tsev algebra over a field of characteristic 0, generated by nil-elements of index 2, is locally solvable. Supported by RFFR grant No. 96-01-01511. Translated fromAlgebra i Logika, Vol. 37, No. 3, pp. 358–373, May–June, 1998.     

Simple Filippov superalgebras of type B(m, n)

This account is a first step toward a classification of finite-dimensional simple Filippov superalgebras over an algebraically closed field of characteristic 0. Here, n-ary Filippov superalgebras with nonzero even and odd parts are treated for the case n ⩾ 3. Supported by RFBR (grant No. 05-01-00230) and by SB RAS (Integration project No. 1.9 and Young Researchers Support grant No. 29). __________ Translated from Algebra i Logika, Vol. 47, No. 2, pp. 240–261, March–April, 2008.     

Generating triples of involutions for lie-type groups over a finite field of odd characteristic. I

It is proved that a simple Lie-type group of rank l≤4 over a field of odd characteristic is generated by three involutions of which two are commuting. As a consequence, the following results obtains: G is generated by two elements one of which is an involution and the order of the other is at most 2h, where h is the Coxeter number of a root system associated with G. Supported by RFFR grant No. 94-01-01084. Translated fromAlgebra i Logika, Vol. 36, No. 1, pp. 77–96, January–February, 1997.     

Vectorial Resilient PC（l） of Order k Boolean Functions from AG-Codes

Propagation criteria and resiliency of vectorial Boolean functions are important for cryptographic purpose (see [1–4, 7, 8, 10, 11, 16]). Kurosawa, Stoh [8] and Carlet [1] gave a construction of Boolean functions satisfying PC(l) of order k from binary linear or nonlinear codes. In this paper, the algebraic-geometric codes over GF(2m) are used to modify the Carlet and Kurosawa-Satoh’s construction for giving vectorial resilient Boolean functions satisfying PC(l) of order k criterion. This new construction is compared with previously known results.     

A generalization of the Razmyslov-Procesi theorem

It is proved that all relations between the invariants of several n x n-matrices over an infinite field of arbitrary characteristic follow from σ_n+1,σ_n+2,... where σ_i is the ith coefficient of a characteristic polynomial extended to matrices of any order ≥i. Similarly, all relations between the concomitants are implied by X_n+1, X_n+2, …, where X_i is a characteristic polynomial in the general n x n-matrix, also extended to matrices of any order. Supported by RFFR grant No. 95-01-00513. Translated fromAlgebra i Logika, Vol. 35, No. 4, pp. 433–457, July–August, 1996.     

Algebraic semantics for superintuitionistic predicate logics

We construct a unified algebraic semantics for superintuitionistic predicate logics. Assigned to each predicate logic is some deductive system of a propositional language which is kept fixed throughout all predicate superintuitionistic ones. Given that system, we build up a variety of algebras w.r.t. which a given logic is proved to be strongly complete. Supported by the Russian Arts Foundation (RAF), grant No. 97-03-04089a. Translated fromAlgebra i Logika, Vol. 38, No. 1, pp. 68–95, January–February, 1999.     

Degrees of presentability of structures. II

We show that the property of being locally constructivizable is inherited under Muchnik reducibility, which is weakest among the effective reducibilities considered over countable structures. It is stated that local constructivizability of level higher than 1 is inherited under Σ-reducibility but is not inherited under Medvedev reducibility. An example of a structure and a relation P ⊆ M is constructed for which but ≢_∑ . Also, we point out a class of structures which are effectively defined by a family of their local theories. Supported by RFBR (grant Nos. 05-0100481 and 06-0104002), by the Council for Grants (under RF President) for State Support of Young Candidates of Science and Their Supervisors (project MK-1239.2005.1), and by INTAS (project YSF 04-83-3310). __________ Translated from Algebra i Logika, Vol. 47, No. 1, pp. 108–126, January–February, 2008.     

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.