Authors:W. Guo; D. O. Revin Pages: 169 - 181 Abstract: Let ð”› be a class of finite groups closed under taking subgroups, homomorphic images, and extensions. Following H. Wielandt, we call a subgroup H of a finite group G a submaximal ð”›-subgroup if there exists an isomorpic embedding ϕ: G ↪ G* of the group G into some finite group G* under which Gϕ is subnormal in G* and Hϕ = K ∩Gϕ for some maximal ð”›-subgroup K of G*. We discuss the following question formulated by Wielandt: Is it always the case that all submaximal ð”›-subgroups are conjugate in a finite group G in which all maximal ð”›-subgroups are conjugate? This question strengthens Wielandt’s known problem of closedness for the class of -groups under extensions, which was solved some time ago. We prove that it is sufficient to answer the question mentioned for the case where G is a simple group. PubDate: 2018-07-01 DOI: 10.1007/s10469-018-9490-9 Issue No:Vol. 57, No. 3 (2018)

Authors:S. S. Korobkov Pages: 186 - 200 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 a lattice isomorphism) of the ring R onto the ring R′. A ring R′ is called the projective image of a ring R. We study lattice isomorphisms of finite commutative rings with identity. The objective is to specify sufficient conditions subject to which rings under lattice homomorphisms preserve the following properties: to be a commutative ring, to be a ring with identity, to be decomposable into a direct sum of ideals. We look into the question about the projective image of the Jacobson radical of a ring. In the first part, the previously obtained results on projections of finite commutative semiprime rings are supplemented with new information. Lattice isomorphisms of finite commutative rings decomposable into direct sums of fields and nilpotent ideals are taken up in the second part. Rings definable by their subring lattices are exemplified. Projections of finite commutative rings decomposable into direct sums of Galois rings and nilpotent ideals are considered in the third part. It is proved that the presence in a ring of a direct summand definable by its subring lattice (i.e., the Galois ring GR(pn,m), where n > 1 and m > 1) leads to strong connections between the properties of R and R′. PubDate: 2018-07-01 DOI: 10.1007/s10469-018-9492-7 Issue No:Vol. 57, No. 3 (2018)

Authors:D. V. Lytkina; V. D. Mazurov Pages: 201 - 210 Abstract: Let G be a periodic group containing an element of order 2 such that each of its finite subgroups of even order lies in a finite subgroup isomorphic to a simple symplectic group of degree 4. It is shown that G is isomorphic to a simple symplectic group S4(Q) of degree 4 over some locally finite field Q. PubDate: 2018-07-01 DOI: 10.1007/s10469-018-9493-6 Issue No:Vol. 57, No. 3 (2018)

Authors:M. P. Shushpanov Pages: 237 - 247 Abstract: It is known that a modular 3-generated lattice is always finite and contains at most 28 elements. Lattices generated by three elements with certain modularity properties may no longer be modular but nevertheless remain finite. It is shown that a 3-generated lattice among generating elements of which one is seminormal and another is coseminormal is finite and contains at most 45 elements. This estimate is stated to be sharp. PubDate: 2018-07-01 DOI: 10.1007/s10469-018-9496-3 Issue No:Vol. 57, No. 3 (2018)

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