Abstract: Let B be a class of groups A which are soluble, equationally Noetherian, and have a central series A = A1 ⩾ A2 ⩾ … An ⩾ … such that ⋂An = 1 and all factors An/An+1 are torsion-free groups; D is a direct product of finitely many cyclic groups of infinite or prime orders. We prove that the wreath product D ≀ A is an equationally Noetherian group. As a consequence we show that free soluble groups of arbitrary derived lengths and ranks are equationally Noetherian.

Abstract: We show that the property of being locally constructivizable is inherited under Muchnik reducibility, which is weakest among the effective reducibilities considered over countable structures. It is stated that local constructivizability of level higher than 1 is inherited under Σ-reducibility but is not inherited under Medvedev reducibility. An example of a structure $$\mathfrak{M}$$ and a relation P ⊆ M is constructed for which $${\underline {({\mathfrak{M}},P)}} \equiv {\underline {\mathfrak{M}}} $$ but $$(\mathfrak{M},P)$$ ≢∑ $$\mathfrak{M}$$ . Also, we point out a class of structures which are effectively defined by a family of their local theories.

Abstract: We argue for the existence of structures with the spectrum {x : x ≰ a} of degrees, where a is an arbitrary low degree. Also it is stated that there exist structures with the spectrum of degrees, {x : x ≰ a} ⋃ {x : x ≰ b}, for any low degrees a and b.

Abstract: We introduce and study some natural operations on a structure of finite labeled forests, which is crucial in extending the difference hierarchy to the case of partitions. It is shown that the corresponding quotient algebra modulo the so-called h-equivalence is the simplest non-trivial semilattice with discrete closures. The algebra is also characterized as a free algebra in some quasivariety. Part of the results is generalized to countable labeled forests with finite chains.

Abstract: We introduce the concept of Hochschild cohomologies for associative conformal algebras. It is shown that the second cohomology group of a conformal Weyl algebra with values in any bimodule is trivial. As a consequence, we derive that the conformal Weyl algebra is segregated in any extension with nilpotent kernel.

Abstract: A non-nilpotent finite group whose proper subgroups are all nilpotent is called a Schmidt group. A subgroup A is said to be seminormal in a group G if there exists a subgroup B such that G = AB and AB1 is a proper subgroup of G, for every proper subgroup B1 of B. Groups that contain seminormal Schmidt subgroups of even order are considered. In particular, we prove that a finite group is solvable if all Schmidt {2, 3}-subgroups and all 5-closed {2, 5}-Schmidt subgroups of the group are seminormal; the classification of finite groups is not used in so doing. Examples of groups are furnished which show that no one of the requirements imposed on the groups is unnecessary.

Abstract: A syntactic approach is described to constructing generic models which generalizes the known semantic one. A sufficient condition of a generic model being homogeneous is specified. It is shown that, within the syntactic approach, any countable homogeneous model is generic. Criteria and a sufficient condition are given for the generic models created in syntactic constructions to be saturated.

Abstract: The interpolation property in extensions of Johansson’s minimal logic is investigated. The construction of a matched product of models is proposed, which allows us to prove the interpolation property in a number of known extensions of the minimal logic. It is shown that, unlike superintuitionistic, positive, and negative logics, a sum of J-logics with the interpolation property CIP may fail to possess CIP, nor even the restricted interpolation property.

Abstract: An involution i of a group G is said to be almost perfect in G if any two involutions of iG the order of a product of which is infinite are conjugated via a suitable involution in iG. We generalize a known result by Brauer, Suzuki, and Wall concerning the structure of finite groups with elementary Abelian centralizers of involutions to groups with almost perfect involutions.

Abstract: Centers of universal envelopes for Mal’tsev algebras are explored. It is proved that the center of the universal envelope for a finite-dimensional semisimple Mal’tsev algebra over a field of characteristic 0 is a ring of polynomials in a finite number of variables equal to the dimension of its Cartan subalgebra, and that universal enveloping algebra is a free module over its center. Centers of universal enveloping algebras are computed for some Mal’tsev algebras of small dimensions.

Abstract: We deal with Sylvan’s logic CCω. It is proved that this logic is a conservative extension of positive intuitionistic logic. Moreover, a paraconsistent extension of Sylvan’s logic is constructed, which is also a conservative extension of positive intuitionistic logic and has the property of being decidable. The constructed logic, in which negation is defined via a total accessibility relation, is a natural intuitionistic analog of the modal system S5. For this logic, an axiomatization is given and the completeness theorem is proved.

Abstract: We fix a universal algebra A and its subalgebra H. The dominion of H in A (in a class M) is the set of all elements a ? A such that any pair of homomorphisms f, g: A ? M ? M satisfies the following: if f and g coincide on H then f(a) = g(a). In association with every quasivariety, therefore, is a dominion of H in A. Sufficient conditions are specified under which a set of dominions form a lattice. The lattice of dominions is explored for down-semidistributivity. We point out a class of algebras (including groups, rings) such that every quasivariety in this class contains an algebra whose lattice of dominions is anti-isomorphic to a lattice of subquasivarieties of that quasivariety.

Abstract: It is proved that in any finite representation of any finitely generated nilpotent group of nilpotency class l ⩾ 1, the averaged Dehn function σ(n) is subasymptotic w.r.t. the function nl+1. As a consequence it is stated that in every finite representation of a free nilpotent group of nilpotency class l of finite rank r ⩾ 2, the Dehn function σ(n) is Gromov subasymptotic.

Abstract: The research launched in [1] is brought to a close by examining algebraic sets in a metabelian group G in two important cases: (1) G = Fn is a free metabelian group of rank n; (2) G = Wn,k is a wreath product of free Abelian groups of ranks n and k.

Abstract: A complete description of (p, 1)-stable theories is furnished in terms of definable interpretability in a theory for the language of unary predicates.

Abstract: Let \(\mathfrak{M}\) be a set of finite groups. A group G is saturated with groups from \(\mathfrak{M}\) if every finite subgroup of G is contained in a subgroup isomorphic to some member of \(\mathfrak{M}\). It is proved that a periodic group G saturated with groups from the set {L3(2m)|m = 1, 2, …} is isomorphic to L3(Q), for a locally finite field Q of characteristic 2; in particular, it is locally finite.

Abstract: We construct an example of a theory with a finite (greater than one) number of isomorphism types of countable models such that its prime and saturated models have computable presentations and there exists a model which lacks in such.

Abstract: We prove that there is a first-order sentence ϕ such that the group of all computable automorphisms of the ordering of the rational numbers is its only model among the groups that are embeddable in the group of all computable permutations.

Abstract: We deal with some upper semilattices of m-degrees and of numberings of finite families. It is proved that the semilattice of all c.e. m-degrees, from which the greatest element is removed, is isomorphic to the semilattice of simple m-degrees, the semilattice of hypersimple m-degrees, and the semilattice of Σ20-computable numberings of a finite family of Σ20-sets, which contains more than one element and does not contain elements that are comparable w.r.t. inclusion.

Abstract: We show that treating of (non-trivial) pairs of irreducible characters of the group Sn sharing the same set of roots on one of the sets An and Sn \ An is divided into three parts. This, in particular, implies that any pair of such characters ?a and ?s (a and s are respective partitions of a number n) possesses the following property: lengths d(a) and d(s) of principal diagonals of Young diagrams for a and s differ by at most 1.

Abstract: The paper deals in questions on the complexity of isomorphisms and relations on universes of structures, on the number and properties of numberings in various hierarchies of sets, and on the existence of connections between semantic and syntactic properties of the structures and relations for the class of nilpotent groups of class two.

Abstract: We prove that if the existence of a supercompact cardinal is consistent with ZFC, then it is consistent with ZFC that the p-rank of Ext ℤ(G, ℤ) is as large as possible for every prime p and for any torsion-free Abelian group G. Moreover, given an uncountable strong limit cardinal µ of countable cofinality and a partition of Π (the set of primes) into two disjoint subsets Π0 and Π1, we show that in some model which is very close to ZFC, there is an almost free Abelian group G of size 2µ = µ+ such that the p-rank of Ext ℤ(G, ℤ) equals 2µ = µ+ for every p ∈ Π0 and 0 otherwise, that is, for p ∈ Π1.

Abstract: We are concerned with infinite-dimensional locally soluble linear groups of infinite central dimension that are not soluble A3-groups and all of whose proper subgroups, which are not soluble A3-groups, have finite central dimension. The structure of groups in this class is described. The case of infinite-dimensional locally nilpotent linear groups satisfying the specified conditions is treated separately. A similar problem is solved for infinite-dimensional locally soluble linear groups of infinite fundamental dimension that are not soluble A3-groups and all of whose proper subgroups, which are not soluble A3-groups, have finite fundamental dimension.

Abstract: We deal with some issues on automatic recognition of interpolation properties in modal calculi extending the logics S5 and S4.3.

Abstract: Let Lq(qG) be the quasivariety lattice contained in a quasivariety generated by a group G. It is proved that if G is a finitely generated torsion-free group in \(\mathcal{A}\mathcal{B}_{2^n } \) (i.e., G is an extension of an Abelian group by a group of exponent 2n), which is a split extension of an Abelian group by a cyclic group, then the lattice Lq(qG) is a finite chain.

Abstract: A subset K of a group G is said to be twisted if 1 ∈ K and xy−1x ∈ K for any x, y ∈ K. We explore finite twisted subsets with involutions which are themselves not subgroups but every proper twisted subset of which is. Groups that are generated by such twisted subsets are classified.

Abstract: We are concerned with a class of valued fields, called stable. We propound an extension of a notion in the monograph by S. Bosch, U. Guntzer, and R. Remmert (Non-Archimedean Analysis. A Systematic Approach to Rigid Analytic Geometry, Springer, Berlin (1984)), namely, that of a (ultrametric) norm on groups, rings, algebras, and vector spaces, to the case where the value of the norm is taken from an arbitrary (not necessarily Archimedean) linearly ordered Abelian group (using — as in the general theory of valuations — the version of a logarithmic norm). Our main result extends Proposition 6 in the cited monograph to the general case, thereby making it possible to use the technique of Cartesian spaces to deliver further results on stable valued fields.

Abstract: Various classes of non-associative algebras possessing the property of being rigid under abstract isomorphisms are studied.

Abstract: The concept of a composition algebra of the second kind is introduced. We prove that such algebras are non-degenerate monocomposition algebras without unity. A big number of these algebras in any finite dimension are constructed, as well as two algebras in a countable dimension. The constructed algebras each contains a non-isotropic idempotent e2 = e. We describe all orthogonally non-isomorphic composition algebras of the second kind in the following forms: (1) a two-dimensional algebra (which has turned out to be unique); (2) three-dimensional algebras in the constructed series. For every algebra A, the group Ortaut A of orthogonal automorphisms is specified.