Abstract: The question about the structure of lattices of subclasses of various classes of algebras is one of the basic ones in universal algebra. The case under consideration most frequently concerns lattices of subvarieties (subquasivarieties) of varieties (quasivarieties) of universal algebras. A similar question is also meaningful for other classes of algebras, in particular, for universal (i.e., axiomatizable by ∀-formulas) classes of algebras. The union of two ∀-classes is itself a ∀-class, hence such lattices are distributive. As a rule, those lattices of subclasses are rather large and are not simply structured. In this connection, it is of interest to distinguish some sublattices of such lattices that would model certain properties of the lattices themselves. The present paper deals with a similar problem for ∀-classes and varieties of universal algebras. PubDate: 2019-11-07

Abstract: Here we give counterexamples to two conjectures in The Kourovka Notebook, Questions 12.78 and 19.67; http://www.math.nsc.ru/∼alglog/19tkt.pdf. The first conjecture concerns character theory of finite groups, and the second one regards permutation group theory. PubDate: 2019-11-07

Abstract: We study the computable reducibility ≤c for equivalence relations in the Ershov hierarchy. For an arbitrary notation a for a nonzero computable ordinal, it is stated that there exist a \( {\varPi}_a^{-1} \) -universal equivalence relation and a weakly precomplete \( {\varSigma}_a^{-1} \) - universal equivalence relation. We prove that for any \( {\varSigma}_a^{-1} \) equivalence relation E, there is a weakly precomplete \( {\varSigma}_a^{-1} \) equivalence relation F such that E ≤cF. For finite levels \( {\varSigma}_m^{-1} \) in the Ershov hierarchy at which m = 4k +1 or m = 4k +2, it is shown that there exist infinitely many ≤c-degrees containing weakly precomplete, proper \( {\varSigma}_m^{-1} \) equivalence relations. PubDate: 2019-11-07

Abstract: We state the following results: the family of all infinite computably enumerable sets has no computable numbering; the family of all infinite \( {\varPi}_1^1 \) sets has no \( {\varPi}_1^1 \) -computable numbering; the family of all infinite \( {\varSigma}_2^1 \) sets has no \( {\varSigma}_2^1 \) -computable numbering. For k > 2, the existence of a \( {\varSigma}_k^1 \) -computable numbering for the family of all infinite \( {\varSigma}_k^1 \) sets leads to the inconsistency of ZF. PubDate: 2019-11-07

Abstract: The exact value of the centralizer dimension is found for a free polynilpotent group and for a free group in a variety of metabelian groups of nilpotency class at most c. Relations between ∃- and Φ-theories of groups are specified, in which case the concept of centralizer dimension plays an important role. PubDate: 2019-11-07

Abstract: It is proved that there exists a set ℛ of quasivarieties of torsion-free groups which (a) have an ω-independent basis of quasi-identities in the class ð’¦0 of torsion-free groups, (b) do not have an independent basis of quasi-identities in ð’¦0, and (c) the intersection of all quasivarieties in ℛ has an independent quasi-identity basis in ð’¦0. The collection of such sets ℛ has the cardinality of the continuum. PubDate: 2019-11-07

Abstract: Using sets of finitely generated Abelian groups closed under the discrimination operator, we describe principal universal classes ð’¦ within a quasivariety ð”„p, the class of groups whose periodic part is a p-group for a prime p. Also the concept of an algebraically closed group in ð’¦ is introduced, and such groups are classified. PubDate: 2019-11-07

Abstract: We prove that in an arbitrary group, the normal closure of a finite Engel element with Artinian centralizer is a locally nilpotent Cĕrnikov subgroup, thereby generalizing the Baer–Suzuki theorem, Blackburn’s and Shunkov’s theorems. PubDate: 2019-11-07

Abstract: We give a criterion for the countable spectrum to be maximal in small binary quite o-minimal theories of finite convexity rank. PubDate: 2019-07-20

Abstract: Let σ = {σi i ∈ I} be a partition of the set of all primes ℙ and G a finite group. Suppose σ(G) = {σi σi ∩ π(G) ≠ = ∅}. A set ℋ of subgroups of G is called a complete Hall σ-set of G if every nontrivial member of ℋ is a σi-subgroup of G for some i ∈ I and ℋ contains exactly one Hall σi-subgroup of G for every i such that σi ∈ σ(G). A group G is σ-full if G possesses a complete Hall σ-set. A complete Hall σ-set ℋ of G is called a σ-basis of G if every two subgroups A, B ∈ ℋ are permutable, i.e., AB = BA. In this paper, we study properties of finite groups having a σ-basis. It is proved that if G has a σ-basis, then G is generalized σ-soluble, i.e, σ(H/K) ≤ 2 for every chief factor H/K of G. Moreover, it is shown that every complete Hall σ-set of a σ-full group G forms a σ-basis of G iff G is generalized σ-soluble, and for the automorphism group G/CG(H/K) induced by G on any its chief factor H/K, we have σ(G/CG(H/K)) ≤ 2 and also σ(H/K) ⊆ σ(G/CG(H/K)) in the case σ(G/CG(H/K)) = 2. PubDate: 2019-07-20

Abstract: The concept of a generalized wreath product of permutation m-groups is introduced, and it is proved that an m-transitive permutation group embeds into a generalized wreath product of its primitive components. PubDate: 2019-07-20

Abstract: The interpolation problem over Johansson’s minimal logic J is considered. We introduce a series of Johansson algebras, which will be used to prove a number of necessary conditions for a J-logic to possess Craig’s interpolation property (CIP). As a consequence, we deduce that there exist only finitely many finite-layered pre-Heyting algebras with CIP. PubDate: 2019-07-20

Abstract: Functions of the algebra of logic that can be realized by read-once formulas over finite bases are studied. Necessary and sufficient conditions are derived under which functions of the algebra of logic are read-once in pre-elementary bases {−, ·,∨, 0, 1, x1 · . . . · xn ∨ \( {\overline{x}}_1 \) · . . . · \( {\overline{x}}_n \) } and {−, ·,∨, 0, 1, x1(x2 ∨ x3 · . . . · xn) ∨ x2 \( {x}_2{\overline{x}}_3 \) · . . . · \( {\overline{x}}_n \) } where n ≥ 4. This completes the description of classes of read-once functions of the algebra of logic in all pre-elementary bases. PubDate: 2019-07-20

Abstract: It is proved that the property of being a semisimple algebra is preserved under projections (lattice isomorphisms) for locally finite-dimensional Lie algebras over a perfect field of characteristic not equal to 2 and 3, except for the projection of a three-dimensional simple nonsplit algebra. Over fields with the same restrictions, we give a lattice characterization of a three-dimensional simple split Lie algebra and a direct product of a one-dimensional algebra and a three-dimensional simple nonsplit one. PubDate: 2019-07-20

Abstract: Sufficient conditions are specified under which a quasivariety contains continuum many subquasivarieties having an independent quasi-equational basis but for which the quasiequational theory and the finite membership problem are undecidable. A number of applications are presented. PubDate: 2019-07-20

Abstract: A standard way to capture the inherent complexity of the isomorphism type of a countable structure is to consider the set of all Turing degrees relative to which the given structure has a computable isomorphic copy. This set is called the degree spectrum of a structure. Similarly, to characterize the complexity of models of a theory, one may examine the set of all degrees relative to which the theory has a computable model. Such a set of degrees is called the degree spectrum of a theory. We generalize these two notions to arbitrary equivalence relations. For a structure \( \mathcal{A} \) and an equivalence relation E, the degree spectrum DgSp( \( \mathcal{A} \) , E) of \( \mathcal{A} \) relative to E is defined to be the set of all degrees capable of computing a structure \( \mathcal{B} \) that is E-equivalent to \( \mathcal{A} \) . Then the standard degree spectrum of \( \mathcal{A} \) is DgSp( \( \mathcal{A} \) , ≅) and the degree spectrum of the theory of \( \mathcal{A} \) is DgSp( \( \mathcal{A} \) , ≡). We consider the relations \( {\equiv}_{\sum_n} \) ( \( \mathcal{A}{\equiv}_{\sum_n}\mathcal{B} \) iff the Σn theories of \( \mathcal{A} \) and \( \mathcal{B} \) coincide) and study degree spectra with respect to \( {\equiv}_{\sum_n} \) . PubDate: 2019-07-20