It is stated that the second Hochshild cohomology group of the associative conformal algebra Cend1,x with values in any bimodule is trivial. Consequently, the given algebra splits off in every extension with nilpotent kernel.

Numerical characteristics of polynomial identities of left nilpotent algebras are examined. Previously, we came up with a construction which, given an infinite binary word, allowed us to build a two-step left nilpotent algebra with specified properties of the codimension sequence. However, the class of the infinite words used was confined to periodic words and Sturm words. Here the previously proposed approach is generalized to a considerably more general case. It is proved that for any algebra constructed given a binary word with subexponential function of combinatorial complexity, there exists a PI-exponent, and its precise value is computed.

We consider 3-generated lattices among generators of which there are elements of distributive and modular types, and one of the generators is necessarily standard. For each triple of such generators, we answer the question whether a lattice generated by that triple is finite.

Algebras of distributions of binary isolating formulas over a type for quite o-minimal theories with nonmaximal number of countable models are described. It is proved that an isomorphism of these algebras for two 1-types is characterized by the coincidence of convexity ranks and also by simultaneous satisfaction of isolation, quasirationality, or irrationality of those types. It is shown that for quite o-minimal theories with nonmaximum many countable models, every algebra of distributions of binary isolating formulas over a pair of nonweakly orthogonal types is a generalized commutative monoid.

A Levi class L(ℳ) generated by a class ℳ of groups is a class of all groups in which the normal closure of each element belongs to ℳ. It is stated that there exist finite groups G such that a Levi class L(qG), where qG is a quasivariety generated by a group G, has infinite axiomatic rank. This is a solution for [The Kourovka Notebook, Quest. 15.36]. Moreover, it is proved that a Levi class L(ℳ), where ℳ is a quasivariety generated by a relatively free 2-step nilpotent group of exponent ps with a commutator subgroup of order p, p is a prime, p ≠ 2, s ≥ 2, is finitely axiomatizable.

For an arbitrary set A of natural numbers, we prove the following statements: every finite family of A-computable sets containing a least element under inclusion has an Acomputable universal numbering; every infinite A-computable family of total functions has (up to A-equivalence) either one A-computable Friedberg numbering or infinitely many such numberings; every A-computable family of total functions which contains a limit function has no A-computable universal numberings, even with respect to Areducibility.

It is known that a modular 3-generated lattice is always finite and contains at most 28 elements. Lattices generated by three elements with certain modularity properties may no longer be modular but nevertheless remain finite. It is shown that a 3-generated lattice among generating elements of which one is seminormal and another is coseminormal is finite and contains at most 45 elements. This estimate is stated to be sharp.

Let Γ be a distance regular graph with intersection array {35, 32, 1; 1, 4, 35} and let G = Aut(Γ) act transitively on the set of vertices of the graph Γ. It is shown that G is a {2, 3}-group.

We find an example of a computable admissible set whose level of computability is higher than that of the standard model of Peano arithmetic. As a byproduct, we construct a 1-decidable model of an undecidable submodel complete theory.

Generalized computable numberings relative to hyperimmune and high oracles are studied. We give a description of oracles relative to which every finite computable family has a universal computable numbering. Also we present a characterization of the class of oracles relative to which every universal computable numbering of an arbitrary finite family is precomplete, and establish a sufficient condition for universal generalized computable numberings to be precomplete. In addition, we look into the question on limitedness of universal numberings computable relative to high oracles.