Vol 58, No 3 (2019)

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 is a weakly precomplete \( {\varSigma}_a^{-1} \) equivalence relation F such that E ≤_cF. For finite levels \( {\varSigma}_m^{-1} \) in the Ershov hierarchy at which m = 4k +1 or m = 4k +2, it is shown that there exist infinitely many ≤_c-degrees containing weakly precomplete, proper \( {\varSigma}_m^{-1} \) equivalence relations.

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 family of all infinite \( {\varSigma}_k^1 \) sets leads to the inconsistency of ZF.

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 by ∀-formulas) classes of algebras. The union of two ∀-classes is itself a ∀-class, hence such lattices are distributive. As a rule, those lattices of subclasses are rather large and are not simply structured. In this connection, it is of interest to distinguish some sublattices of such lattices that would model certain properties of the lattices themselves. The present paper deals with a similar problem for ∀-classes and varieties of universal algebras.

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.