Showing papers in "Algebra and Logic in 2019"
TL;DR: In this article, the authors find a sufficient condition for a quasivariety K to have many subquasivariates that have no independent quasi-equational bases relative to K, but have ω-independent quasiequational base relative to k. This condition also implies that K is Q-universal.
Abstract: We find a sufficient condition for a quasivariety K to have continuum many subquasivarieties that have no independent quasi-equational bases relative to K but have ω-independent quasi-equational bases relative to K. This condition also implies that K is Q-universal.
15 citations
TL;DR: In this paper, sufficient conditions are specified under which a quasivariety contains many subquasivariates having an independent quasi-equational basis but for which the quasiequational theory and the finite membership problem are undecidable.
Abstract: 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.
11 citations
TL;DR: In this paper, a universal enveloping Lie Rota-Baxter algebras of pre-Lie and post-Lie was constructed, and it was proved that the pairs of varieties (RBLie, preLie) and (RBλLie, postLie) are PBW-pairs.
Abstract: Universal enveloping Lie Rota–Baxter algebras of pre-Lie and post-Lie algebras are constructed. It is proved that the pairs of varieties (RBLie, preLie) and (RBλLie, postLie) are PBW-pairs and that the variety of Lie Rota–Baxter algebras is not a Schreier variety.
6 citations
TL;DR: The family of all infinite computably enumerable sets has no computable numbering, and the existence of a a _k^1 -computable numbering for the family ofall infinite $$ {\varSigma}_k-k-1 $$ sets leads to the inconsistency of ZF.
Abstract: 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.
4 citations
TL;DR: For quite o-minimal theories with non-maximum many countable models, every algebra of distributions of binary isolating formulas over a pair of nonweakly orthogonal types is a generalized commutative monoid.
Abstract: Algebras of distributions of binary isolating formulas over a type for quite o-minimal theories with nonmaximal number of countable models are described. It is proved that an isomorphism of these algebras for two 1-types is characterized by the coincidence of convexity ranks and also by simultaneous satisfaction of isolation, quasirationality, or irrationality of those types. It is shown that for quite o-minimal theories with nonmaximum many countable models, every algebra of distributions of binary isolating formulas over a pair of nonweakly orthogonal types is a generalized commutative monoid.
4 citations
TL;DR: In this article, it was shown that the theory of divisible m-rigid groups admits quantifier elimination down to a Boolean combination of ∃-formulas, which implies that these tuples are conjugate via an automorphism of G.
Abstract: A group G is said to be rigid if it contains a normal series G = G1 > G2 > . . . > Gm > Gm+1 = 1, whose quotients Gi/Gi+1 are Abelian and, treated as right ℤ[G/Gi]-modules, are torsion-free. A rigid group G is divisible if elements of the quotient Gi/Gi+1 are divisible by nonzero elements of the ring ℤ[G/Gi]. Every rigid group is embedded in a divisible one. Our main result is the theorem which reads as follows. Let G be a divisible rigid group. Then the coincidence of ∃-types of same-length tuples of elements of the group G implies that these tuples are conjugate via an automorphism of G. As corollaries we state that divisible rigid groups are strongly ℵ0-homogeneous and that the theory of divisible m-rigid groups admits quantifier elimination down to a Boolean combination of ∃-formulas.
4 citations
TL;DR: In this article, counterexamples to two conjectures in The Kourovka Notebook, Questions 12.78 and 19.67 are given for finite groups and permutation group theory.
Abstract: 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.
4 citations
TL;DR: In this article, it was shown that a simple right-alternative unital superalgebra over a field of characteristic not 2, whose even part coincides with an algebra of matrices of order 2, is isomorphic to an asymmetric double.
Abstract: We classify simple right-alternative unital superalgebras over a field of characteristic not 2, whose even part coincides with an algebra of matrices of order 2. It is proved that such a superalgebra either is a Wall double W2|2(ω), or is a Shestakov superalgebra S4|2(σ) (characteristic 3), or is isomorphic to an asymmetric double, an 8-dimensional superalgebra depending on four parameters. In the case of an algebraically closed base field, every such superalgebra is isomorphic to an associative Wall double M2[√1], an alternative 6-dimensional Shestakov superalgebra B4|2 (characteristic 3), or an 8-dimensional Silva–Murakami–Shestakov superalgebra.
3 citations
3 citations
TL;DR: In this article, the authors give sufficient conditions for generalized computable numberings to satisfy the statement of Khutoretskii's theorem, which implies limitedness of universal computable numbers for 2 = \le \alpha <{\omega}_1^{CK}.
Abstract: We give sufficient conditions for generalized computable numberings to satisfy the statement of Khutoretskii’s theorem. This implies limitedness of universal $$ {\varSigma}_{\alpha}^0- $$ computable numberings for 2 $$ \le \alpha <{\omega}_1^{CK}. $$
3 citations
TL;DR: In this article, it was shown that the second Hochshild cohomology group of the associative conformal algebra Cend1,x with values in any bimodule is trivial.
Abstract: It is stated that the second Hochshild cohomology group of the associative conformal algebra Cend1,x with values in any bimodule is trivial. Consequently, the given algebra splits off in every extension with nilpotent kernel.
TL;DR: In this paper, the connection between the notion of a subset defined in the language of ℳ over ω and a syntactic complexity class was studied. And the connections among the notions of √ √ 1/0, √ 2/0 and √ d-Sigma were studied.
Abstract: 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.
TL;DR: In this paper, it was shown that there exists a set ℛ of quasivarieties of torsion-free groups which have an ω-independent basis of quasi-identities in the class 𝒦0 of TGFs.
Abstract: 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 𝒦0 of torsion-free groups, (b) do not have an independent basis of quasi-identities in 𝒦0, and (c) the intersection of all quasivarieties in ℛ has an independent quasi-identity basis in 𝒦0. The collection of such sets ℛ has the cardinality of the continuum.
TL;DR: For finite simple groups U5(2n), n > 1, U4(q), and S 4(q) as discussed by the authors, the minimum number of generating conjugate involutions whose product equals 1 is equal to 5.
Abstract: For finite simple groups U5(2n), n > 1, U4(q), and S4(q), where q is a power of a prime p > 2, q − 1 ≠= 0(mod4), and q ≠= 3, we explicitly specify generating triples of involutions two of which commute. As a corollary, it is inferred that for the given simple groups, the minimum number of generating conjugate involutions, whose product equals 1, is equal to 5.
TL;DR: In this article, the authors give a criterion for the countable spectrum to be maximal in small binary quite o-minimal theories of finite convexity rank, and prove that it is maximal in the case where the theory is a quite ominimal theory.
Abstract: We give a criterion for the countable spectrum to be maximal in small binary quite o-minimal theories of finite convexity rank.
TL;DR: It is shown that there exist infinitely many ≤c-degrees containing weakly precomplete, proper {\varSigma}_m^{-1} $$ equivalence relations.
Abstract: We study the computable reducibility ≤c for equivalence relations in the Ershov hierarchy. For an arbitrary notation a for a nonzero computable ordinal, it is stated that there exist a $$ {\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} $$ equivalence relations.
TL;DR: The degree spectrum of a theory is generalized to arbitrary equivalence relations iff the Σn theories of A and B coincide and study degree spectra with respect to ≡∑n$$ {\equiv}_{\sum_n} $$.
Abstract: A standard way to capture the inherent complexity of the isomorphism type of a countable structure is to consider the set of all Turing degrees relative to which the given structure has a computable isomorphic copy. This set is called the degree spectrum of a structure. Similarly, to characterize the complexity of models of a theory, one may examine the set of all degrees relative to which the theory has a computable model. Such a set of degrees is called the degree spectrum of a theory. We generalize these two notions to arbitrary equivalence relations. For a structure $$ \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} $$
.
TL;DR: It is shown that every structure can be transformed into a graph that is bi-interpretable with the original structure, for which the full elementary diagrams can be computed one from the other.
Abstract: It is shown that every structure (including one in an infinite language) can be transformed into a graph that is bi-interpretable with the original structure, for which the full elementary diagrams can be computed one from the other.
TL;DR: In this paper, it was shown that the normal closure of a finite Engel element with Artinian centralizer is a locally nilpotent C-rnikov subgroup, thereby generalizing the Baer-Suzuki theorem, Blackburn's and Shunkov's theorems.
Abstract: We prove that in an arbitrary group, the normal closure of a finite Engel element with Artinian centralizer is a locally nilpotent Cĕrnikov subgroup, thereby generalizing the Baer–Suzuki theorem, Blackburn’s and Shunkov’s theorems.
TL;DR: In this article, the exact value of the centralizer dimension for a free polynilpotent group and a free group in a variety of metabelian groups of nilpotency class at most c.
Abstract: The exact value of the centralizer dimension is found for a free polynilpotent group and for a free group in a variety of metabelian groups of nilpotency class at most c. Relations between ∃- and Φ-theories of groups are specified, in which case the concept of centralizer dimension plays an important role.
TL;DR: It was shown in this article that the ordinal ω1cannot be embedded into a preordering Σ-definable with parameters in the hereditarily finite superstructure over the real numbers.
Abstract: It is proved that the ordinal ω1cannot be embedded into a preordering Σ-definable with parameters in the hereditarily finite superstructure over the real numbers. As a corollary, we obtain the descriptions of ordinals Σ-presentable over$$ \mathbb{H}\mathbbm{F} $$(ℝ) and of Godel constructive sets of the form Lα. It is also shown that there are no Σ-presentations of structures of T-, m-, 1- and tt-degrees.
TL;DR: In this article, it was shown that for any algebra constructed given a binary word with subexponential function of combinatorial complexity, there exists a PI-exponent, and its precise value was computed.
Abstract: Numerical characteristics of polynomial identities of left nilpotent algebras are examined. Previously, we came up with a construction which, given an infinite binary word, allowed us to build a two-step left nilpotent algebra with specified properties of the codimension sequence. However, the class of the infinite words used was confined to periodic words and Sturm words. Here the previously proposed approach is generalized to a considerably more general case. It is proved that for any algebra constructed given a binary word with subexponential function of combinatorial complexity, there exists a PI-exponent. And its precise value is computed.
TL;DR: In this article, the problem of finding dis-limits for a given algebraic structure is formulated as a set of classes and quasivarieties, i.e., algebraic structures in which all irreducible coordinate algebras over a field are embedded and in which there are no other finitely generated substructures.
Abstract: This paper enters into a series of works on universal algebraic geometry—a branch of mathematics that is presently flourishing and is still undergoing active development. The theme and subject area of universal algebraic geometry have their origins in classical algebraic geometry over a field, while the language and almost the entire methodological apparatus belong to model theory and universal algebra. The focus of the paper is the problem of finding Dis-limits for a given algebraic structure $$ \mathcal{A} $$
, i.e., algebraic structures in which all irreducible coordinate algebras over $$ \mathcal{A} $$
are embedded and in which there are no other finitely generated substructures. Finding a solution to this problem necessitated a good description of principal universal classes and quasivarieties. The paper is divided into two parts. In the first part, we give criteria for a given universal class (or quasivariety) to be principal. In the second part, we formulate explicitly the problem of finding Dis-limits for algebraic structures and show how the results of the first part make it possible to solve this problem in many cases.
TL;DR: In this article, the lattice definability of a matrix ring over an arbitrary Galois ring is studied and a complete description of projective images of prime and semiprime finite rings is given.
Abstract: Associative rings R and R′ are said to be lattice-isomorphic if their subring lattices L(R) and L(R′) are isomorphic. An isomorphism of the lattice L(R) onto the lattice L(R′) is called a projection (or lattice isomorphism) of the ring R onto the ring R′. A ring R′ is called the projective image of a ring R. Whenever a lattice isomorphism φ implies an isomorphism between R and Rφ, we say that the ring R is determined by its subring lattice. The present paper is a continuation of previous research on lattice isomorphisms of finite rings. We give a complete description of projective images of prime and semiprime finite rings. One of the basic results is the theorem on lattice definability of a matrix ring over an arbitrary Galois ring. Projective images of finite rings decomposable into direct sums of matrix rings over Galois rings of different types are described.
TL;DR: The interpolation problem over Johansson's minimal logic J is considered in this paper, where a series of Johansson algebras are used to prove a number of necessary conditions for a J-logic to possess CIP.
Abstract: 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.
TL;DR: In this paper, the authors consider 3-generated lattices among generators of which there are elements of distributive and modular types, and answer the question whether a lattice generated by that triple is finite.
Abstract: We consider 3-generated lattices among generators of which there are elements of distributive and modular types, and one of the generators is necessarily standard. For each triple of such generators, we answer the question whether a lattice generated by that triple is finite.
TL;DR: In this paper, it was shown that for an arbitrary monoid S, the class of divisible polygons is primitive normal if and only if S is a linearly ordered monoid, and that it is primitive connected if F is a group.
Abstract: 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.
TL;DR: In this paper, it was shown that the adjacency matrix of Cayley graphs (G, S) is an integer, and that all eigenvalues of Cay(G, s) are integers.
Abstract: Let G be a group and S ⊆ G a subset such that S = S−1, where S−1 = {s−1 | s ∈ S}. Then a Cayley graph Cay(G, S) is an undirected graph Γ with vertex set V (Γ) = G and edge set E(Γ) = {(g, gs) | g ∈ G, s ∈ S}. For a normal subset S of a finite group G such that s ∈ S ⇒ sk ∈ S for every k ∈ ℤ which is coprime to the order of s, we prove that all eigenvalues of the adjacency matrix of Cay(G, S) are integers. Using this fact, we give affirmative answers to Questions 19.50(a) and 19.50(b) in the Kourovka Notebook.