Abstract: We discuss how meaningful is the concept of an epimaximal Ӿ -subgroup dual to the concept of a submaximal Ӿ -subgroup introduced by H. Wielandt. Also a result of Wielandt is refined which characterizes the behavior of maximal Ӿ -subgroups under homomorphisms. PubDate: 2020-03-03

Abstract: Classifications of logics over Johansson’s minimal logic J and modal logics are considered. The paper contains a partial review of the results obtained after 2010. It is known that there is a duality between the lattice of normal logics and the lattice of varieties of modal algebras, as well as between the lattice of varieties of J-algebras and the lattice of J-logics. For a logic L, by V (L) we denote its corresponding variety of algebras. PubDate: 2020-03-03

Abstract: We introduce the construction of a generalized direct product associated with a graph of groups and prove two sufficient conditions for its existence. These results are applied to obtain some sufficient conditions for an HNN-extension with central associated subgroups to be residually a C-group where C is a root class of groups. In particular, it is proved that an HNN-extension of a solvable group with central associated subgroups is residually solvable. PubDate: 2020-03-03

Abstract: We give a sufficient condition for a quasivariety K, weaker than the one found earlier by A. V. Kravchenko, A. M. Nurakunov, and the author, which ensures that K contains continuum many subquasivarieties with no independent quasi-equational basis relative to K. This condition holds, in particular, for any almost f f-universal quasivariety K. PubDate: 2020-03-03

Abstract: We show that there is a one-to-one correspondence (up to isomorphism) between commutative rings with unity and metabelian commutative loops belonging to a particular finitely axiomatizable class. Based on this correspondence, it is proved that the sets of identically valid formulas and of finitely refutable formulas of a class of finite nonassociative commutative loops (and of many of its other subclasses) are recursively inseparable. It is also stated that nonassociative commutative free automorphic loops of any nilpotency class have an undecidable elementary theory. PubDate: 2020-03-03

Abstract: It is proved that the field of complex algebraic numbers has an isomorphic presentation computable in polynomial time. A similar fact is proved for the ordered field of real algebraic numbers. The constructed polynomially computable presentations are based on a natural presentation of algebraic numbers by rational polynomials. Also new algorithms for computing values of polynomials on algebraic numbers and for solving equations in one variable with algebraic coefficients are presented. PubDate: 2020-03-03

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: 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

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: 2019-09-01