Authors:N. M. Suchkov Abstract: We prove the following theorem. Let U be a locally finite Suzuki–Higman 2-group with respect to an automorphism group H. Then U and H are representable as the respective unions of ascending chains of finite subgroups U1 < U2 < . . . < U n < . . . and H1 < H2 < . . . < H n < . . ., in which case every subgroup U n is a Suzuki 2-group with respect to H n . PubDate: 2018-02-26 DOI: 10.1007/s10469-018-9470-0

Authors:D. O. Ptakhov Abstract: Polygons with a (P, 1)-stable theory are considered. A criterion of being (P, 1)-stable for a polygon is established. As a consequence of the main criterion we prove that a polygon SS, where S is a group, is (P, 1)-stable if and only if S is a finite group. It is shown that the class of all polygons with monoid S is (P, 1)-stable only if S is a one-element monoid. (P, 1)-stability criteria are presented for polygons over right and left zero monoids. PubDate: 2018-02-24 DOI: 10.1007/s10469-018-9469-6

Authors:E. P. Vdovin; M. N. Nesterov; D. O. Revin Abstract: It is known that for any set π of prime numbers, the following assertions are equivalent: (1) in any finite group, π-Hall subgroups are conjugate; (2) in any finite group, π-Hall subgroups are pronormal. It is proved that (1) and (2) are equivalent also to the following: (3) in any finite group, π-Hall subgroups are pronormal in their normal closure. Previously [10, Quest. 18.32], the question was posed whether it is true that in a finite group, π-Hall subgroups are always pronormal in their normal closure. Recently, M. N. Nesterov [7] proved that assertion (3) and assertions (1) and (2) are equivalent for any finite set π. The fact that there exist examples of finite sets π and finite groups G such that G contains more than one conjugacy class of π-Hall subgroups gives a negative answer to the question mentioned. Our main result shows that the requirement of finiteness for π is unessential for (1), (2), and (3) to be equivalent. PubDate: 2018-02-24 DOI: 10.1007/s10469-018-9467-8

Authors:A. S. Morozov Abstract: It is proved that any countable consistent theory with infinite models has a Σ-presentable model of cardinality 2ω over ℍð”½(ℝ). It is shown that some structures studied in analysis (in particular, a semigroup of continuous functions, certain structures of nonstandard analysis, and infinite-dimensional separable Hilbert spaces) have no simple Σ-presentations in hereditarily finite superstructures over existentially Steinitz structures. The results are proved by a unified method on the basis of a new general sufficient condition. PubDate: 2018-02-24 DOI: 10.1007/s10469-018-9468-7

Authors:V. V. Bitkina; A. A. Makhnev Abstract: Let Γ be a distance regular graph with intersection array {35, 32, 1; 1, 4, 35} and let G = Aut(Γ) act transitively on the set of vertices of the graph Γ. It is shown that G is a {2, 3}-group. PubDate: 2018-02-24 DOI: 10.1007/s10469-018-9466-9

Authors:P. E. Alaev Abstract: We consider a new approach to investigating categoricity of structures computable in polynomial time. The approach is based on studying polynomially computable stable relations. It is shown that this categoricity is equivalent to the usual computable categoricity for computable Boolean algebras with computable set of atoms, and for computable linear orderings with computable set of adjacent pairs. Examples are constructed which show that this does not always hold. We establish a connection between dimensions based on computable and polynomially computable stable relations. PubDate: 2018-02-24 DOI: 10.1007/s10469-018-9465-x

Authors:V. G. Bardakov; M. V. Neshchadim Pages: 355 - 361 Abstract: We study a representation of the virtual braid group VBn into the automorphism group of a free product of a free group and a free Abelian group, proposed by S. Kamada. It is proved that the given representation is equivalent to the representation constructed in [http://arxiv.org/abs/1603.01425]; i.e. the kernels of these representations coincide. PubDate: 2017-11-01 DOI: 10.1007/s10469-017-9457-2 Issue No:Vol. 56, No. 5 (2017)

Authors:A. V. Kislitsin Pages: 362 - 369 Abstract: Conditions are explored which imply the finite basis property for identities of vector spaces embedded in associative algebras over an infinite field. An L-variety having no finite basis of identities, which is the join of two Spechtian L-varieties, is exemplified. PubDate: 2017-11-01 DOI: 10.1007/s10469-017-9458-1 Issue No:Vol. 56, No. 5 (2017)

Authors:L. L. Maksimova; V. F. Yun Pages: 370 - 385 Abstract: Extensions of Johansson’s minimal logic J are considered. It is proved that families of negative and nontrivial logics and a series of other families are strongly decidable over J. This means that, given any finite list Rul of axiom schemes and rules of inference, we can effectively verify whether the logic with axioms and schemes, J + Rul, belongs to a given family. Strong recognizability over J is proved for known logics Neg, Gl, and KC as well as for logics LC and NC and all their extensions. PubDate: 2017-11-01 DOI: 10.1007/s10469-017-9459-0 Issue No:Vol. 56, No. 5 (2017)

Authors:I. A. Mal’tsev Pages: 386 - 394 Abstract: We consider the possibility for separating by hyperidentities clones of quasilinear functions defined on the set {0, 1, 2} with values in the set {0, 1}. It is proved that every creative clone of this kind can be separated by a hyperidentity from any noncreative clone comparable with it. PubDate: 2017-11-01 DOI: 10.1007/s10469-017-9460-7 Issue No:Vol. 56, No. 5 (2017)

Authors:N. S. Romanovskii Pages: 395 - 408 Abstract: A group G is said to be rigid if it contains a normal series G = G 1 > G 2 > … > Gm > G m+1 = 1, whose quotients G i/G i+1 are Abelian and, treated as right ℤ[G/G i ]-modules, are torsion-free. A rigid group G is divisible if elements of the quotient G i /G i+1 are divisible by nonzero elements of the ring ℤ[G/G i ]. Every rigid group is embedded in a divisible one. We prove two theorems. Theorem 1 says that the following three conditions for a group G are equivalent: G is algebraically closed in the class Σ m of all m-rigid groups; G is existentially closed in the class Σ m ; G is a divisible m-rigid group. Theorem 2 states that the elementary theory of a class of divisible m-rigid groups is complete. PubDate: 2017-11-01 DOI: 10.1007/s10469-017-9461-6 Issue No:Vol. 56, No. 5 (2017)

Authors:M. V. Schwidefsky Pages: 409 - 424 Abstract: We give a characterization of complete strongly dually atomic lattices having unique irredundant decompositions which are also canonical. It is shown that all known characterizations of lattices with unique irredundant decompositions are a consequence of this result. In addition, upper continuous closure lattices of convex geometries with (unique) irredundant decompositions are characterized. PubDate: 2017-11-01 DOI: 10.1007/s10469-017-9462-5 Issue No:Vol. 56, No. 5 (2017)

Authors:A. A. Stepanova; D. O. Ptakhov Abstract: P-stable polygons are studied. It is proved that the property of being (P, s)-, (P, a)-, and (P, e)-stable for the class of all polygons over a monoid S is equivalent to S being a group. We describe the structure of (P, s)-, (P, a)-, and (P, e)-stable polygons SA over a countable left zero monoid S and, under the condition that the set A \ SA is indiscernible, over a right zero monoid. PubDate: 2017-11-07 DOI: 10.1007/s10469-017-9453-6

Authors:R. A. Kornev Abstract: We study computable reducibility of computable metrics on R induced by reducibility of their respective Cauchy representations. It is proved that this ordering has a subordering isomorphic to an arbitrary countable tree. Also we introduce a weak version of computable reducibility and construct a countable antichain of computable metrics that are incomparable with respect to it. Informally, copies of the real line equipped with these metrics are pairwise homeomorphic but not computably homeomorphic. PubDate: 2017-11-07 DOI: 10.1007/s10469-017-9451-8