Algebra and Logic
This journal publishes results of the latest research in the areas of modern general algebra and of logic considered primarily from an algebraic viewpoint. The algebraic papers, constituting the major part of the contents, are concerned with studies in such fields as ordered, almost torsion-free, nilpotent, and metabelian groups; isomorphism rings; Lie algebras; Frattini subgroups; and clusters of algebras. In the area of logic, the periodical covers such topics as hierarchical sets, logical automata, and recursive functions.
Algebra and Logic is a translation of the peer-reviewed journal Algebra I Logika, a publication of the Siberian Fund for Algebra and Logic and the Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences.
More information is available at the editor's website via the following link:
http://math.nsc.ru/~alglog/
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Vol 57, No 6 (2019)
- Year: 2019
- Articles: 7
- URL: https://journals.rcsi.science/0002-5232/issue/view/14552
Article
Free 3-Generated Lattices with Standard Element Among Generators
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.
399-413
Algebraic Geometry Over Algebraic Structures. IX. Principal Universal Classes and Dis-Limits
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.
414-428
Algebras of Distributions of Binary Isolating Formulas for Quite o-Minimal Theories
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.
429-444
Structure of Quasivariety Lattices. I. Independent Axiomatizability
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.
445-462
A Combinatorial Classification of Finite Quasigroups
Abstract
For a finite groupoid with right cancellation, we define the concepts of a bicycle, of a bicyclic decomposition, and of a bicyclic action of the symmetric group of permutations on a groupoid. An isomorphism criterion based on a bicyclic decomposition gives rise to an effective method for solving problems such as establishing an isomorphism between finite groups with right cancellation, finding their automorphism groups, and listing their subgroupoids. We define an operation of the square of a groupoid using its bicyclic decomposition, which allows one to recognize a quasigroup in a groupoid with right cancellation. On a set of n-element quasigroups, we introduce the equivalent relations of being isomorphic and of being of a single type. The factor set of the single-type relation is ordered by an order type relation consistent with squares of quasigroups. A set of n-element quasigroups is representable as a union of nonintersecting sequences of quasigroups ordered by a relation of comparison of types of single-type classes that contain them.
463-477
Divisible Rigid Groups. III. Homogeneity and Quantifier Elimination
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.
478-489
Sessions of the Seminar “Algebra i Logika”
490-491