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Том 58, № 2 (2019)
- Год: 2019
- Статей: 9
- URL: https://journal-vniispk.ru/0002-5232/issue/view/14554
Article
Projections of Semisimple Lie Algebras
Аннотация
It is proved that the property of being a semisimple algebra is preserved under projections (lattice isomorphisms) for locally finite-dimensional Lie algebras over a perfect field of characteristic not equal to 2 and 3, except for the projection of a three-dimensional simple nonsplit algebra. Over fields with the same restrictions, we give a lattice characterization of a three-dimensional simple split Lie algebra and a direct product of a one-dimensional algebra and a three-dimensional simple nonsplit one.
103-114
115-122
Structure of Quasivariety Lattices. II. Undecidable Problems
Аннотация
Sufficient conditions are specified under which a quasivariety contains continuum many subquasivarieties having an independent quasi-equational basis but for which the quasiequational theory and the finite membership problem are undecidable. A number of applications are presented.
123-136
137-143
The Interpolation Problem in Finite-Layered Pre-Heyting Logics
Аннотация
The interpolation problem over Johansson’s minimal logic J is considered. We introduce a series of Johansson algebras, which will be used to prove a number of necessary conditions for a J-logic to possess Craig’s interpolation property (CIP). As a consequence, we deduce that there exist only finitely many finite-layered pre-Heyting algebras with CIP.
144-157
Degree Spectra of Structures Relative to Equivalences
Аннотация
A standard way to capture the inherent complexity of the isomorphism type of a countable structure is to consider the set of all Turing degrees relative to which the given structure has a computable isomorphic copy. This set is called the degree spectrum of a structure. Similarly, to characterize the complexity of models of a theory, one may examine the set of all degrees relative to which the theory has a computable model. Such a set of degrees is called the degree spectrum of a theory. We generalize these two notions to arbitrary equivalence relations. For a structure \( \mathcal{A} \) and an equivalence relation E, the degree spectrum DgSp(\( \mathcal{A} \), E) of \( \mathcal{A} \) relative to E is defined to be the set of all degrees capable of computing a structure \( \mathcal{B} \) that is E-equivalent to \( \mathcal{A} \). Then the standard degree spectrum of \( \mathcal{A} \) is DgSp(\( \mathcal{A} \), ≅) and the degree spectrum of the theory of \( \mathcal{A} \) is DgSp(\( \mathcal{A} \), ≡). We consider the relations \( {\equiv}_{\sum_n} \) (\( \mathcal{A}{\equiv}_{\sum_n}\mathcal{B} \) iff the Σn theories of \( \mathcal{A} \) and \( \mathcal{B} \) coincide) and study degree spectra with respect to \( {\equiv}_{\sum_n} \).
158-172
Finite Generalized Soluble Groups
Аннотация
Let σ = {σi | i ∈ I} be a partition of the set of all primes ℙ and G a finite group. Suppose σ(G) = {σi | σi ∩ π(G) ≠ = ∅}. A set ℋ of subgroups of G is called a complete Hall σ-set of G if every nontrivial member of ℋ is a σi-subgroup of G for some i ∈ I and ℋ contains exactly one Hall σi-subgroup of G for every i such that σi ∈ σ(G). A group G is σ-full if G possesses a complete Hall σ-set. A complete Hall σ-set ℋ of G is called a σ-basis of G if every two subgroups A, B ∈ ℋ are permutable, i.e., AB = BA. In this paper, we study properties of finite groups having a σ-basis. It is proved that if G has a σ-basis, then G is generalized σ-soluble, i.e, |σ(H/K)| ≤ 2 for every chief factor H/K of G. Moreover, it is shown that every complete Hall σ-set of a σ-full group G forms a σ-basis of G iff G is generalized σ-soluble, and for the automorphism group G/CG(H/K) induced by G on any its chief factor H/K, we have |σ(G/CG(H/K))| ≤ 2 and also σ(H/K) ⊆ σ(G/CG(H/K)) in the case |σ(G/CG(H/K))| = 2.
173-185
Read-Once Functions of the Algebra of Logic in Pre-Elementary Bases
Аннотация
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.
186-195
Sessions of the Seminar “Algebra i Logika”
196-198