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

Abstract: We formulate a polynomial completeness criterion for quasigroups of prime order, and show that verification of polynomial completeness may require time polynomial in order. The results obtained are generalized to n-quasigroups for any n ≥ 3. In conclusion, simple corollaries are given on the share of polynomially complete quasigroups among all quasigroups, and on the cycle structure of row and column permutations in Cayley tables for quasigroups that are not polynomially complete. PubDate: 2018-12-01 DOI: 10.1007/s10469-018-9505-6

Abstract: Nearly finite-dimensional Jordan algebras are examined. Analogs of known results are considered. Namely, it is proved that such algebras are prime and nondegenerate. It is shown that the property of being nearly finite-dimensional is preserved in passing from an alternative algebra to an adjoint Jordan algebra. A similar result is established for associative nearly finite-dimensional algebras with involution. It is stated that a nearly finite-dimensional Jordan PI-algebra with unity either is a finite module over a nearly finite-dimensional center or is a central order in an algebra of a nondegenerate symmetric bilinear form. Also the following result holds: if a locally nilpotent ideal has finite codimension in a Jordan algebra with the ascending chain condition on ideals, then that algebra is finite-dimensional. In addition, E. Formanek’s result in [Comm. Alg., 1, No. 1, 79-86 (1974)], which says that associative prime PI-rings with unity are embedded in a free module of finite rank over its center, is generalized to Albert rings. PubDate: 2018-12-01 DOI: 10.1007/s10469-018-9506-5

Authors:A. L. Kanunnikov Abstract: We specify conditions on a group G that are necessary and sufficient for analogs of Goldie’s theorems to hold in a class of G-graded rings, i.e., for every G-graded gr-prime (gr-semiprime) right Goldie ring to possess a completely gr-reducible graded classical right ring of quotients. PubDate: 2018-11-30 DOI: 10.1007/s10469-018-9507-4

Authors:I. Sh. Kalimullin; V. G. Puzarenko; M. Kh. Faizrakhmanov Abstract: The objects considered here serve both as generalizations of numberings studied in [1] and as particular versions of A-numberings, where ð”¸ is a suitable admissible set, introduced in [2] (in view of the existence of a transformation realizing the passage from e-degrees to admissible sets [3]). The key problem dealt with in the present paper is the existence of Friedberg (single-valued computable) and positive presentations of families. In [3], it was stated that the above-mentioned transformation preserves the majority of properties treated in descriptive set theory. However, it is not hard to show that it also respects the positive (negative, decidable, single-valued) presentations. Note that we will have to extend the concept of a numbering and, in the general case, consider partial maps rather than total ones. The given effect arises under the passage from a hereditarily finite superstructure to natural numbers, since a computable function (in the sense of a hereditarily finite superstructure) realizing an enumeration of the hereditarily finite superstructure for nontotal sets is necessarily a partial function. PubDate: 2018-11-21 DOI: 10.1007/s10469-018-9503-8

Authors:A. N. Rybalov Abstract: Generic amplification is a method that allows algebraically undecidable problems to generate problems undecidable for almost all inputs. It is proved that every simple negligible set is undecidable for almost all inputs, but it cannot be obtained via amplification from any undecidable set. On the other hand, it is shown that every recursively enumerable set with nonzero asymptotic density can be obtained via amplification from a set of natural numbers. PubDate: 2018-11-21 DOI: 10.1007/s10469-018-9500-y

Authors:S. A. Badaev; A. A. Issakhov Abstract: For an arbitrary set A of natural numbers, we prove the following statements: every finite family of A-computable sets containing a least element under inclusion has an Acomputable universal numbering; every infinite A-computable family of total functions has (up to A-equivalence) either one A-computable Friedberg numbering or infinitely many such numberings; every A-computable family of total functions which contains a limit function has no A-computable universal numberings, even with respect to Areducibility. PubDate: 2018-11-21 DOI: 10.1007/s10469-018-9499-0

Authors:A. N. Khisamiev Abstract: It is proved that a universal Σ-function exists in a hereditarily finite superstructure over an unbounded branching tree of finite height. PubDate: 2018-11-21 DOI: 10.1007/s10469-018-9502-9

Authors:E. I. Timoshenko Abstract: We study elementary and universal theories of relatively free solvable groups in a group signature expanded by one predicate distinguishing primitive or annihilating systems of elements. PubDate: 2018-11-21 DOI: 10.1007/s10469-018-9501-x

Authors:P. E. Alaev Abstract: We define a class \( \mathbb{K} \) Σ of primitive recursive structures whose existential diagram is decidable with primitive recursive witnesses. It is proved that a Boolean algebra has a presentation in \( \mathbb{K} \) Σ iff it has a computable presentation with computable set of atoms. Moreover, such a Boolean algebra is primitive recursively categorical with respect to \( \mathbb{K} \) Σ iff it has finitely many atoms. The obtained results can also be carried over to Boolean algebras computable in polynomial time. PubDate: 2018-11-20 DOI: 10.1007/s10469-018-9498-1