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Том 58, № 3 (2019)
- Год: 2019
- Статей: 11
- URL: https://journal-vniispk.ru/0002-5232/issue/view/14555
Article
Weakly Precomplete Equivalence Relations in the Ershov Hierarchy
Аннотация
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.
199-213
ω-Independent Bases for Quasivarieites of Torsion-Free Groups
Аннотация
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 ????0 of torsion-free groups, (b) do not have an independent basis of quasi-identities in ????0, and (c) the intersection of all quasivarieties in ℛ has an independent quasi-identity basis in ????0. The collection of such sets ℛ has the cardinality of the continuum.
214-223
Computable Numberings of Families of Infinite Sets
Аннотация
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.
224-231
Canonical and Algebraically Closed Groups in Universal Classes of Abelian Groups
Аннотация
Using sets of finitely generated Abelian groups closed under the discrimination operator, we describe principal universal classes ???? within a quasivariety ????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 ???? is introduced, and such groups are classified.
232-243
Lattices of Boundedly Axiomatizable ∀-Subclasses of ∀-Classes of Universal Algebras
Аннотация
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.
244-248
Counterexamples to Two Conjectures in the Kourovka Notebook
Аннотация
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.
249-253
254-267
Universal Theories and Centralizer Dimensions of Groups
Аннотация
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.
268-281
Generating Sets of Involutions of Finite Simple Groups
288-293
Sessions of the Seminar “Algebra i Logika”
294-295
Communications
Turing Degrees of Complete Formulas of Almost Prime Models
282-287