We point out an existence criterion for positive computable total \( {\Pi}_1^1 \) -numberings of families of subsets of a given \( {\Pi}_1^1 \) -set. In particular, it is stated that the family of all \( {\Pi}_1^1 \) -sets has no positive computable total \( {\Pi}_1^1 \) -numberings. Also we obtain a criterion of existence for computable Friedberg \( {\Sigma}_1^1 \) -numberings of families of subsets

We discuss a well-known conjecture that the full automorphism group of a finite projective plane coordinatized by a semifield is solvable. For a semifield plane of order pN (p > 2 is a prime, 4|p − 1) admitting an autotopism subgroup H isomorphic to the quaternion group Q8, we construct a matrix representation of H and a regular set of the plane. All nonisomorphic semifield planes of orders 54 and

We study an extension of temporal logic, a multi-agent logic on models with nontransitive linear time (which is, in a sense, also an extension of interval logic). The proposed relational models admit lacunas in admissibility relations among agents: information accessible for one agent may be inaccessible for others. A logical language uses temporary operators ‘until’ and ‘next’ (for each of the agents)

It is proved that there exist uncountably many nonisomorphic pro-p-groups admitting presentations with two generating elements and one defining relation in the variety of metabelian pro-p-groups. Also we give information on pro-p-groups with a single relation in varieties of nilpotent pro-p-groups.

We specify normalizers of Sylow r-subgroups in finite simple linear and unitary groups for the case where r is an odd prime distinct from the characteristic of a definition field of a group.

The study of automorphisms of computable and other structures connects computability theory with classical group theory. Among the noncomputable countable structures, computably enumerable structures are one of the most important objects of investigation in computable model theory. Here we focus on the lattice structure of computably enumerable substructures of a given canonical computable structure

Associative rings are considered. By a lattice isomorphism, or projection, of a ring R onto a ring Rφ we mean an isomorphism φ of the subring lattice L(R) of R onto the subring lattice L(Rφ) of Rφ. In this case Rφ is called the projective image of a ring R and R is called the projective preimage of a ring Rφ. Let R be a finite ring with identity and Rad R the Jacobson radical of R. A ring R is said

We deal with questions concerning the completeness and stability of a class of injective acts and a class of weakly injective acts over a monoid S. The concepts of an injective S-act and of a weakly injective S-act are analogs of the concept of an injective module. In the theory of modules, the corresponding notions of injectivities in accordance with Baer’s criterion coincide. Also we will look into

It is proved that the d-rank of an arbitrary α-space does not exceed 1.

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.

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

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

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.

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)

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

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

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.

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.

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

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.

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

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

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.