Abstract: Let L(M) be a class of all groups G in which the normal closure of any element belongs to M; qM is a quasivariety generated by a class M. We consider a quasivariety qH2 generated by a relatively free group in a class of nilpotent groups of class at most 2 with commutator subgroup of exponent 2. It is proved that the Levi class L(qH2) generated by the quasivariety qH2 is contained in the variety of nilpotent groups of class at most 3. PubDate: 2019-11-18

Abstract: It is proved that in a unital alternative algebra A of characteristic ≠ 2, the associator (a, b, c) and the Kleinfeld function f(a, b, c, d) never assume the value 1 for any elements a, b, c, d ∈ A. Moreover, if A is nonassociative, then no commutator [a, b] can be equal to 1. As a consequence, there do not exist algebraically closed alternative algebras. The restriction on the characteristic is essential, as exemplified by the Cayley–Dickson algebra over a field of characteristic 2. PubDate: 2019-11-18

Abstract: Let ℳ be a structure of a signature Σ. For any ordered tuple \( \overline{a}=\left({a}_1,\dots, {a}_{\mathrm{n}}\right) \) of elements of ℳ, \( {\mathrm{tp}}^{\mathcal{M}}\left(\overline{a}\right) \) denotes the set of formulas θ(x1, …, xn) of a first-order language over Σ with free variables x1, . . . , xn such that \( \mathcal{M}\left =\theta \left({a}_1,\dots, {a}_n\right)\right. \). A structure ℳ is said to be strongly ω-homogeneous if, for any finite ordered tuples \( \overline{a} \) and \( \overline{b} \) of elements of ℳ, the coincidence of \( {\mathrm{tp}}^{\mathcal{M}}\left(\overline{a}\right) \) and \( {\mathrm{tp}}^{\mathrm{M}}\left(\overline{b}\right) \) implies that these tuples are mapped into each other (componentwise) by some automorphism of the structure ℳ. A structure ℳ is said to be prime in its theory if it is elementarily embedded in every structure of the theory Th (ℳ). It is proved that the integral group rings of finitely generated relatively free orderable groups are prime in their theories, and that this property is shared by the following finitely generated countable structures: free nilpotent associative rings and algebras, free nilpotent rings and Lie algebras. It is also shown that finitely generated non-Abelian free nilpotent associative algebras and finitely generated non-Abelian free nilpotent Lie algebras over uncountable fields are strongly ω-homogeneous. PubDate: 2019-11-18

Abstract: We give sufficient conditions for generalized computable numberings to satisfy the statement of Khutoretskii’s theorem. This implies limitedness of universal \( {\varSigma}_{\alpha}^0- \) computable numberings for 2 \( \le \alpha <{\omega}_1^{CK}. \) PubDate: 2019-11-18

Abstract: Let A be a numerical k ×∞-matrix such that minors AI of order k tend to zero if numbers of all columns forming these minors tend to infinity. It is shown that there exits a nontrivial linear combination of rows in A which is a sequence tending to zero. PubDate: 2019-11-18

Abstract: It is proved that the universal equivalence of general or special linear groups of orders greater than 2 over local commutative rings with 1/2 is equivalent to the coincidence of orders of groups and universal equivalence of respective rings. PubDate: 2019-11-16

Abstract: We describe a maximal variety ð”š of automorphic Moufang loops such that for every loop A in the variety ð”š, any loop isotopic to A also lies in ð”š. PubDate: 2019-11-16

Abstract: Let G be a group and S ⊆ G a subset such that S = S−1, where S−1 = {s−1 s ∈ S}. Then a Cayley graph Cay(G, S) is an undirected graph Γ with vertex set V (Γ) = G and edge set E(Γ) = {(g, gs) g ∈ G, s ∈ S}. For a normal subset S of a finite group G such that s ∈ S ⇒ sk ∈ S for every k ∈ ℤ which is coprime to the order of s, we prove that all eigenvalues of the adjacency matrix of Cay(G, S) are integers. Using this fact, we give affirmative answers to Questions 19.50(a) and 19.50(b) in the Kourovka Notebook. PubDate: 2019-11-16

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