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Том 55, № 4 (2016)
- Год: 2016
- Статей: 11
- URL: https://journals.rcsi.science/0002-5232/issue/view/14538
Article
257-273
Undecidable Iterative Propositional Calculus
Аннотация
We consider iterative propositional calculi that are finite sets of propositional formulas together with modus ponens and an operation of superposition defined by a set of Mal’tsev operations. For such formulas, the question is studied whether the derivability problem for formulas is decidable. In the paper, we construct an undecidable iterative propositional calculus whose axioms depend on three variables. A derivation of formulas in the given calculus models the solution process for Post’s correspondence problem. In particular, we prove that the general problem of expressibility for iterative propositional calculi is algorithmically undecidable.
274-282
∏11-Completeness of the Computable Categoricity Problem for Projective Planes
Аннотация
Computable presentations for projective planes are studied. We prove that the problem of computable categoricity is ∏11-complete for the following classes of projective planes: Pappian projective planes, Desarguesian projective planes, arbitrary projective planes.
283-288
Periodic Groups Saturated with Finite Simple Groups of Types U3 and L3
Аннотация
Suppose that \( \mathfrak{M} \) is a set whose elements are simple three-dimensional unitary groups U3(q) and linear groups L3(q) over finite fields. We prove that a periodic group saturated with groups of \( \mathfrak{M} \) is locally finite and isomorphic to U3(Q) or L3(Q) for some locally finite field Q.
289-294
Layers over Minimal Logic
Аннотация
We introduce a classification of extensions of Johansson’s minimal logic J that extends the classification of superintuitionistic logics proposed by T. Hosoi. It is proved that the layer number of any finitely axiomatizable logic is effectively computable. Every layer over J has a least logic. It is stated that each layer has finitely many maximal logics, and minimal and maximal logics of all layers are recognizable over J.
295-305
Index Set of Structures with Two Equivalence Relations That Are Autostable Relative to Strong Constructivizations
Аннотация
We derive a bound on the algorithmic complexity for the class of computable structures with two equivalence relations that have a strong constructivization and are autostable relative to strong constructivizations. We construct codings of a linear order and of an automorphically nontrivial directed irreflexive graph into a structure with two equivalence relations. It is proved that such codings preserve the degree spectrum and d-computable dimension.
306-314
Decomposition of a Group over an Abelian Normal Subgroup
Аннотация
Let a group G have an Abelian normal subgroup A; put \( \overline{G} \) = G/A and \( \overline{g} \) = gA for g ∈ G. We can think of A as a right ℤ\( \overline{G} \)-module and define the action of an element \( u={\alpha}_1{\overline{g}}_1 \) +…+ \( {\alpha}_n{\overline{g}}_n \) ∈ ℤ\( \overline{G} \) on a ∈ A by a formula au = \( {\left({a}^{g_1}\right)}^{\alpha_1} \) ·…· \( {\left({a}^{g_n}\right)}^{\alpha_n} \) ; here \( {a}^{g_i}={g}_i^{-1}a{g}_i \). Denote by \( {\Theta}_{\mathbb{Z}\overline{G}} \)(A) the annihilator of A in the ring ℤ\( \overline{G} \), which is a two-sided ideal. Let \( R=\mathbb{Z}\overline{G}/{\Theta}_{\mathbb{Z}\overline{G}}(A) \). A subgroup A can also be treated as an R-module. We give a criterion for the existence of an R-decomposition of G over A, i.e., the possibility of embedding G in a semidirect product \( \overline{G} \)·D, where D is an R-module. It is also proved that an R-decomposition always exists in one important case.
315-326
327-329
330-339
Sessions of the Seminar “Algebra i Logika”
345-346
Communications
On L. G. Kovàcs’ Problem
340-344