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. PubDate: 2019-05-30

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. PubDate: 2019-05-30

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. PubDate: 2019-05-30

Abstract: We study the structure of an infinite group with automorphism of order 2p, where p is an odd prime leaving only the identity element fixed. PubDate: 2019-05-30

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. PubDate: 2019-05-30

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. PubDate: 2019-05-30

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. PubDate: 2019-05-30

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. PubDate: 2019-03-25

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. PubDate: 2019-03-25

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. PubDate: 2019-03-25

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. PubDate: 2019-03-22

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. PubDate: 2019-03-22

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. PubDate: 2019-03-22

Abstract: We study the Specht property for L-varieties of vector spaces embedded in associative algebras over an arbitrary field. An L-variety with no finite basis of identities over a field, which is the join of two Spechtian L-varieties, is exemplified. A condition under which L-varieties will have the Specht property is found. PubDate: 2018-12-01 DOI: 10.1007/s10469-018-9508-3

Abstract: A Levi class L(ℳ) generated by a class ℳ of groups is a class of all groups in which the normal closure of each element belongs to ℳ. It is stated that there exist finite groups G such that a Levi class L(qG), where qG is a quasivariety generated by a group G, has infinite axiomatic rank. This is a solution for [The Kourovka Notebook, Quest. 15.36]. Moreover, it is proved that a Levi class L(ℳ), where ℳ is a quasivariety generated by a relatively free 2-step nilpotent group of exponent ps with a commutator subgroup of order p, p is a prime, p ≠ 2, s ≥ 2, is finitely axiomatizable. PubDate: 2018-12-01 DOI: 10.1007/s10469-018-9510-9

Abstract: We come up with a semantic method of forcing formulas by finite structures in an arbitrary fixed Fraïssé class . Both known and some new necessary and sufficient conditions are derived under which a given structure will be a forcing structure. A formula φ is forced on \( \overline{a} \) in an infinite structure ╟φ \( \left(\overline{a}\right) \) if it is forced in by some finite substructure of . It is proved that every ∃∀∃-sentence true in a forcing structure is also true in any existentially closed companion of the structure. The new concept of a forcing type plays an important role in studying forcing models. It is proved that an arbitrary structure will be a forcing structure iff all existential types realized in the structure are forcing types. It turns out that an existentially closed structure which is simple over a tuple realizing a forcing type will itself be a forcing structure. Moreover, every forcing type is realized in an existentially closed structure that is a model of a complete theory of its forcing companion. PubDate: 2018-12-01 DOI: 10.1007/s10469-018-9509-2