Algebra and Logic最新文献

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.

According to G. Birkhoff, there is a categorical duality between the category of bi-algebraic distributive (0, 1)-lattices with complete (0, 1)-lattice homomorphisms as morphisms and the category of partially ordered sets with partial order-preserving maps as morphisms. We extend this classical result to the bi-algebraic lattices belonging to the variety of (0, 1)-lattices generated by the pentagon, the 5-element nonmodular lattice. Applying the extended duality, we prove that the lattice of quasivarieties contained in the variety of (0, 1)-lattices generated by the pentagon has uncountably many elements and is not distributive. This yields the following: the lattice of quasivarieties contained in a nontrivial variety of (0, 1)-lattices either is a 2-element chain or has uncountably many elements and is not distributive.

We generalize the constructions of Q- and G-families of quandles introduced in the paper of A. Ishii et al. in [Ill. J. Math., 57, No. 3, 817-838 (2013)], and establish how they are related to other constructions of quandles. A composition of structures of quandles defined on the same set is specified, and conditions are found under which this composition yields a quandle. It is proved that under such a multiplication we obtain a group that will be Abelian. Also a direct product of quandles is examined.