A finitely generated group acting on a tree so that all vertex and edge stabilizers are infinite cyclic groups is called a generalized Baumslag–Solitar group (a GBS group). Every GBS group is the fundamental group π1(𝔸) of a suitable labeled graph 𝔸. We prove that if 𝔸 and 𝔹 are labeled trees, then the groups π1(𝔸) and π1(𝔹) are universally equivalent iff π1(𝔸) and π1(𝔹) are embeddable into

We give a full description of associative algebras over an arbitrary field, whose subalgebra lattice is distributive. All such algebras are commutative, their nil-radical is at most two-dimensional, and the factor algebra with respect to the nil-radical is an algebraic extension of the base field.

The notion of general recursive realizability is defined based on using indices of general recursive functions as a constructive way of obtaining some realizations from others. The soundness of basic logic with respect to the semantics of general recursive realizability is proved.

Automorphisms of a graph with intersection array {nm − 1, nm− n + m − 1, n − m + 1; 1, 1, nm− n + m − 1} are considered.

Abelian groups are described over which the class of all subdirectly irreducible acts is axiomatizable. Also some properties of subdirectly irreducible acts over Abelian groups are studied. It is proved that all connected acts over an Abelian group are subdirectly irreducible iff the group is totally ordered.

We point out possible automorphisms of a distance-regular graph Γ with intersection array {55, 54, 2; 1, 1, 54} and spectrum 551, 71617,−1110,−81408.

It is proved that for any nilpotent subgroups A and B in a finite group G with sporadic socle, there is an element g such that A ∩ Bg = 1.

We look at Baumslag–Solitar groups. Lower central series are given for residually nilpotent Baumslag–Solitar groups. For some Baumslag–Solitar groups that are not residually nilpotent, we find the intersection of all terms of the lower central series. Also we point out non-Abelian Baumslag–Solitar groups having lower central series of length 2. For some Baumslag–Solitar groups, a connection is found

We consider algebras of binary formulas for compositions of theories both in the general case and as applied to ℵ0-categorical, strongly minimal, and stable theories, linear preorders, cyclic preorders, and series of finite structures. It is shown that edefinable compositions preserve isomorphisms and elementary equivalence and have basicity formed by basic formulas of the initial theories. We find

We consider two kinds of reducibilities on finite families of predicates on a countable set: the definability of predicates and their complements of one family via another by means of existential formulas with parameters and the same definability on isomorphism types of families. Ordered structures of degrees generated by families of unary predicates are described. It is proved that for both reducibilities

In a previous paper, on a collection FA of functional clones on a set A, we introduced a natural metric d turning it into a topological (metric) space \( {\mathfrak{F}}_A=\left\langle {F}_A;d\right\rangle . \) In this paper, we describe the structure of neighborhoods of clones in spaces \( {\mathfrak{F}}_A \) and establish a number of consequences of this result.

A finite group G is called a generalized Frobenius group with kernel F if F is a proper nontrivial normal subgroup of G, and for every element Fx of prime order p in the quotient group G/F, the coset Fx of G consists of p-elements. We study generalized Frobenius groups with an insoluble kernel F. It is proved that F has a unique non- Abelian composition factor, and that this factor is isomorphic to

We prove that each quasivariety containing a B-class has continuum many subquasivarieties with finitely partitionable ω-independent quasi-equational basis.

We study the structure Ceprs induced by degrees of computably enumerable preorder relations with respect to computable reducibility ≤c. It is proved that the structure of computably enumerable equivalence relations is definable in Ceprs. This fact and results of Andrews, Schweber, and Sorbi imply that the theory of the structure Ceprs is computably isomorphic to first-order arithmetic. It is shown

We establish a series of results on verbally closed and l-verbally closed subgroups of free solvable groups; here l is a natural number and the concept of l-verbal closedness is a generalization of the concept of verbal closedness corresponding to the value l = 1. Under certain assumptions, these subgroups turn out to be retracts and, consequently, are algebraically closed.

A group G is said to be rigid if it contains a normal series G = G1 > G2 > … > Gm > Gm+1 = 1, whose quotients Gi/Gi+1 are Abelian and, when treated as right ℤ[G/Gi]-modules, are torsion-free. A rigid group G is divisible if elements of the quotient Gi/Gi+1 are divisible by nonzero elements of the ring ℤ[G/Gi]. We describe subgroups of a divisible rigid group which are definable in the signature of

An identity of degree 6 is given which is satisfied in an alternative monster and in a Skosyrskii algebra. It is proved that this identity is not a consequence of the alternativity identity, the commutativity identity, and an identity of nil-index 3. Also it is the only identity of degree at most 6, which is not a consequence of the identities mentioned. Therefore, the alternative monster and the Skosyrskii

Maximal solvable subgroup subgroup of odd index. Maximal solvable subgroups of odd index in symmetric groups are classified up to conjugation.

Automorphisms of a partially commutative metabelian group whose defining graph contains no cycles are studied. It is proved that an IA-automorphism of such a group is identical if it fixes all hanging and isolated vertices of the graph. The concepts of a factor automorphism and of a matrix automorphism are introduced. It is stated that every factor automorphism is represented as the product of an automorphism

The Lambek calculus (a variant of intuitionistic linear logic initially introduced for mathematical linguistics) enjoys natural interpretations over the algebra of formal languages (L-models) and over the algebra of binary relations which are subsets of a given transitive relation (R-models). For both classes of models there are completeness theorems (Andréka and Mikulás [J. Logic Lang. Inf., 3, No

The paper deals monoids over which the class of all injective S-acts is primitive normal and primitive connected. The following results are proved: the class of all injective acts over any monoid is primitive normal; the class of all injective acts over a right reversible monoid S is primitive connected iff S is a group; if a monoid S is not a group and the class of all injective acts is primitive

We look at the structure of a unital right alternative superalgebra of capacity 1 over an algebraically closed field assuming that its even part is finite-dimensional and strongly alternative. It is proved that the condition of being simple for such a superalgebra implies the simplicity of its even part.

The name of the author should read not A. V. Vasil’ev, but A. S. Vasil’ev.

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.

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 \( \le \alpha <{\omega}_1^{CK}. \)

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( x 1, …,  x n ) of a first-order language over Σ with free variables x 1 , . . . , x n such that \( \mathcal{M}\left|=\theta \left({a}_1,\dots, {a}_n\right)\right. \). A structure

Let A be a numerical k ×∞-matrix such that minors A I 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.

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

We describe a maximal variety 𝔚 of automorphic Moufang loops such that for every loop A in the variety 𝔚, any loop isotopic to A also lies in 𝔚.

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 ⇒ s k ∈ S for every k ∈ ℤ which is coprime to the order of s, we prove that all eigenvalues of the adjacency matrix of Cay(

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.

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.

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.

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.

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

Using sets of finitely generated Abelian groups closed under the discrimination operator, we describe principal universal classes 𝒦 within a quasivariety 𝔄 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 𝒦 is introduced, and such groups are classified.

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

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 𝒦0 of torsion-free groups, (b) do not have an independent basis of quasi-identities in 𝒦0, and (c) the intersection of all quasivarieties in ℛ has an independent quasi-identity basis in 𝒦0. The collection of such sets ℛ has the cardinality of the

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

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, x 1 · . . . · x n ∨ \( {\overline{x}}_1 \)· . . . · \( {\overline{x}}_n \)} and {−, ·,∨, 0, 1, x 1( x 2 ∨ x 3 · . . . · x n ) ∨ x 2\( {x}_2{\overline{x}}_3