Algebra and Logic最新文献

A finitely generated group that acts on a tree so that all vertex and edge stabilizers are infinite cyclic groups is called a generalized Baumslag–Solitar group (GBS group). Every GBS group is the fundamental group π₁(𝔸) of a suitable labeled graph 𝔸. If 𝔸 is a labeled tree, then we call π₁(𝔸) a tree GBS group. It is known that GBS groups isomorphic to tree groups are themselves tree groups. It is shown that GBS groups universally equivalent to tree groups do not have to be tree groups. Moreover, we give a description of all GBS groups that are universally equivalent to tree groups.

We prove the theorem stating the following. Let G be a locally finite group saturated with groups from a set 𝔐 consisting of direct products of d dihedral groups. Then G is a direct product of d groups of the form B ⋋ <υ>, where B is a locally cyclic group inverted by an involution υ.

We study countable models of Ehrenfeucht theories, i.e., complete theories with a finite number of countable models, strictly larger than 1. The notion of a primely generated model is introduced. It is proved that if all complete types of an Ehrenfeucht theory have arithmetic complexity, then any of the primely generated models of the theory possesses an arithmetically complex isomorphic presentation.

Let 𝔛 be a nonempty class of finite groups closed under taking subgroups, homomorphic images, and extensions. We define the concept of a Wielandt 𝔛 -subgroup in an arbitrary finite group. It generalizes the concept of a submaximal 𝔛 -subgroup introduced by H. Wielandt and is key in the framework of a program proposed by Wielandt in 1979. One of the central objectives of the program is to overcome difficulties associated with the reduction to factors of a subnormal series within the natural problem of searching for maximal 𝔛 -subgroups. Wielandt 𝔛 -subgroups possess a number of properties unshareable by submaximal 𝔛 -subgroups. There is a hope that, due to these additional properties, the use of Wielandt 𝔛 -subgroups will open up new possibilities in realizing Wielandt’s program.

We consider the relationship between the CEA-hierarchy and the Ershov hierarchy in ({Delta }_{2}^{0}) Turing degrees. A degree c is called CEA(a) if c is computably enumerable in a, and a ≤ c. Soare and Stob [Stud. Logic Found. Math., 107, 299-324 (1982)] proved that for a noncomputable low c.e. degree a there exists a CEA(a) degree that is not c.e. Later, Arslanov, Lempp, and Shore [Ann. Pure Appl. Logic, 78, Nos. 1-3, 29-56 (1996)] formulated the problem of describing pairs of degrees a < e such that there exists a CEA(a) 2-c.e. degree d ≤ e which is not c.e. Since then the question has remained open as to whether a CEA(a) degree in the sense of Soare and Stob can be made 2-c.e. Here we answer this question in the negative, solving it in a stronger formulation: there exists a noncomputable low c.e. degree a such that any CEA(a) ω-c.e. degree is c.e. Also possible generalizations of the result obtained are discussed, as well as various issues associated with the problem mentioned.

We start studying the structure of pseudofinite unars. Necessary (sufficient) conditions of being pseudofinite are formulated for connected unars without cycles, containing no chains, and we give examples showing that these conditions are not sufficient (resp. necessary). It is noted that a coproduct of chains is a pseudofinite unar; in particular, a chain is a pseudofinite unar. A nonpseudofinite unar without cycles, containing exactly one chain is exemplified. For connected unars without cycles, containing two chains, we formulate a necessary condition of being pseudofinite and give an example of a nonpseudofinite unar.

We look into the question about conditions under which any permutation of an indiscernible subset of a model extends to an automorphism of the model.

An algebraic, model-theoretic, and algorithmic theory of enriched Boolean algebras with distinguished ideals was developed in a series of papers by D. E. Pal’chunov, A. Touraille, P. E. Alaev, N. T. Kogabaev, and other authors. Here we study the problem on the number of countable Boolean algebras with distinguished ideals for the case when an algebra and its quotient with respect to a distinguished ideal are atomless. It is proved that, for this subclass, there exist continuum many such countable structures.