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 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.

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.

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.

Using sets of finitely generated Abelian groups closed under the discrimination operator, we describe principal universal classes 𝒦 within a quasivariety 𝔄p, 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.

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.

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, x1 · . . . · xn ∨ $$ {\overline{x}}_1 $$ · . . . · $$ {\overline{x}}_n $$ } and {−, ·,∨, 0, 1, x1(x2 ∨ x3 · . . . · xn) ∨ x2 $$ {x}_2{\overline{x}}_3 $$ · . . . · $$ {\overline{x}}_n $$ } where n ≥ 4. This completes the description of classes of read-once functions of the algebra of logic in all pre-elementary bases.