Abstract: It is shown that every structure (including one in an infinite language) can be transformed into a graph that is bi-interpretable with the original structure, for which the full elementary diagrams can be computed one from the other. PubDate: 2019-12-03

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 monoids over which a class of divisible S-polygons is primitive normal or primitive connected. It is shown that for an arbitrary monoid S, the class of divisible polygons is primitive normal iff S is a linearly ordered monoid, and that it is primitive connected iff S is a group. PubDate: 2019-11-01

Abstract: Given a structure ℳ over ω and a syntactic complexity class \( \mathfrak{E} \), we say that a subset is \( \mathfrak{E} \)-definable in ℳ if there exists a C-formula Θ(x) in the language of ℳ such that for all x ∈ ω, we have x ∈ A iff Θ(x) is true in the structure. S. S. Goncharov and N. T. Kogabaev [Vestnik NGU, Mat., Mekh., Inf., 8, No. 4, 23-32 (2008)] generalized an idea proposed by Friedberg [J. Symb. Log., 23, No. 3, 309-316 (1958)], introducing the notion of a \( \mathfrak{E} \)-classification of M: a computable list of \( \mathfrak{E} \)-formulas such that every \( \mathfrak{E} \)-definable subset is defined by a unique formula in the list. We study the connections among\( {\varSigma}_1^0- \), \( d-{\varSigma}_1^0- \), and \( {\varSigma}_2^0 \)-classifications in the context of two families of structures, unbounded computable equivalence structures and unbounded computable injection structures. It is stated that every such injection structure has a \( {\varSigma}_1^0- \)classification, a \( {\varSigma}_1^0- \)classification, and a \( {\varSigma}_2^0 \)-classification. In equivalence structures, on the other hand, we find a richer variety of possibilities. PubDate: 2019-11-01

Abstract: A simple right-alternative, but not alternative, superalgebra whose even part coincides with an algebra of second-order matrices is called an asymmetric double. It is known that such superalgebras are eight-dimensional. We give a solution to the isomorphism problem for asymmetric doubles, point out their automorphism groups and derivation superalgebras. PubDate: 2019-11-01

Abstract: It is proved that the ordinal ω1cannot be embedded into a preordering Σ-definable with parameters in the hereditarily finite superstructure over the real numbers. As a corollary, we obtain the descriptions of ordinals Σ-presentable over\( \mathbb{H}\mathbbm{F} \)(ℝ) and of Gödel constructive sets of the form Lα. It is also shown that there are no Σ-presentations of structures of T-, m-, 1- and tt-degrees. PubDate: 2019-11-01