Abstract: An identity of degree 6 is given which is satisfied in an alternative monster and in a Skosyrskii algebra. It is proved that this identity is not a consequence of the alternativity identity, the commutativity identity, and an identity of nil-index 3. Also it is the only identity of degree at most 6, which is not a consequence of the identities mentioned. Therefore, the alternative monster and the Skosyrskii algebra are known to satisfy five independent identities. PubDate: 2020-07-29

Abstract: Maximal solvable subgroup subgroup of odd index. Maximal solvable subgroups of odd index in symmetric groups are classified up to conjugation. PubDate: 2020-07-29

Abstract: Automorphisms of a partially commutative metabelian group whose defining graph contains no cycles are studied. It is proved that an IA-automorphism of such a group is identical if it fixes all hanging and isolated vertices of the graph. The concepts of a factor automorphism and of a matrix automorphism are introduced. It is stated that every factor automorphism is represented as the product of an automorphism of the defining graph and a matrix automorphism. PubDate: 2020-07-29

Abstract: The Lambek calculus (a variant of intuitionistic linear logic initially introduced for mathematical linguistics) enjoys natural interpretations over the algebra of formal languages (L-models) and over the algebra of binary relations which are subsets of a given transitive relation (R-models). For both classes of models there are completeness theorems (Andréka and Mikulás [J. Logic Lang. Inf., 3, No. 1, 1-37 (1994)]; Pentus [Ann. Pure Appl. Logic, 75, Nos. 1/2, 179-213 (1995); Fund. Prikl. Mat., 5, No. 1, 193- 219 (1999)]). The operations of the Lambek calculus include product and two divisions, left and right. We consider an extension of the Lambek calculus with intersection and iteration (Kleene star). It is proved that this extension is incomplete both w.r.t. L-models and w.r.t. R-models. We introduce a restricted fragment, in which iteration is allowed only in denominators of division operations. For this fragment we prove completeness w.r.t. R-models. We also prove completeness w.r.t. L-models for the subsystem without product. Both results are strong completeness theorems, i.e., they establish equivalence between derivability from sets of hypotheses (finite or infinite) and semantic entailment from sets of hypotheses on a given class of models. Finally, we prove \( {\Pi}_1^0 \) -completeness of the algorithmic problem of derivability in the restricted fragment in question. PubDate: 2020-07-29

Abstract: The paper deals monoids over which the class of all injective S-acts is primitive normal and primitive connected. The following results are proved: the class of all injective acts over any monoid is primitive normal; the class of all injective acts over a right reversible monoid S is primitive connected iff S is a group; if a monoid S is not a group and the class of all injective acts is primitive connected, then a maximal (w.r.t. inclusion) proper subact of SS is not finitely generated. PubDate: 2020-07-29

Abstract: We look at the structure of a unital right alternative superalgebra of capacity 1 over an algebraically closed field assuming that its even part is finite-dimensional and strongly alternative. It is proved that the condition of being simple for such a superalgebra implies the simplicity of its even part. PubDate: 2020-07-29

Abstract: We point out an existence criterion for positive computable total \( {\Pi}_1^1 \) -numberings of families of subsets of a given \( {\Pi}_1^1 \) -set. In particular, it is stated that the family of all \( {\Pi}_1^1 \) -sets has no positive computable total \( {\Pi}_1^1 \) -numberings. Also we obtain a criterion of existence for computable Friedberg \( {\Sigma}_1^1 \) -numberings of families of subsets of a given \( {\Sigma}_1^1 \) - set, the consequence of which is the absence of a computable Friedberg \( {\Sigma}_1^1 \) -numbering of the family of all \( {\Sigma}_1^1 \) -sets. Questions concerning the existence of negative computable \( {\Pi}_1^1 \) - and \( {\Sigma}_1^1 \) -numberings of the families mentioned are considered. PubDate: 2020-05-22

Abstract: We discuss a well-known conjecture that the full automorphism group of a finite projective plane coordinatized by a semifield is solvable. For a semifield plane of order pN (p > 2 is a prime, 4 p − 1) admitting an autotopism subgroup H isomorphic to the quaternion group Q8, we construct a matrix representation of H and a regular set of the plane. All nonisomorphic semifield planes of orders 54 and 134 admitting Q8 in the autotopism group are pointed out. It is proved that a semifield plane of order p4, 4 p−1, does not admit SL(2, 5) in the autotopism group. PubDate: 2020-05-22

Abstract: We study an extension of temporal logic, a multi-agent logic on models with nontransitive linear time (which is, in a sense, also an extension of interval logic). The proposed relational models admit lacunas in admissibility relations among agents: information accessible for one agent may be inaccessible for others. A logical language uses temporary operators ‘until’ and ‘next’ (for each of the agents), via which we can introduce modal operations ‘possible’ and ‘necessary.’ The main problem under study for the logic introduced is the recognition problem for admissibility of inference rules. Previously, this problem was dealt with for a logic in which transitivity intervals have a fixed uniform length. Here the uniformity of length is not assumed, and the logic is extended by individual temporal operators for different agents. An algorithm is found which decides the admissibility problem in a given logic, i.e., it recognizes admissible inference rules. PubDate: 2020-05-22

Abstract: It is proved that there exist uncountably many nonisomorphic pro-p-groups admitting presentations with two generating elements and one defining relation in the variety of metabelian pro-p-groups. Also we give information on pro-p-groups with a single relation in varieties of nilpotent pro-p-groups. PubDate: 2020-05-21

Abstract: We specify normalizers of Sylow r-subgroups in finite simple linear and unitary groups for the case where r is an odd prime distinct from the characteristic of a definition field of a group. PubDate: 2020-05-21

Abstract: The study of automorphisms of computable and other structures connects computability theory with classical group theory. Among the noncomputable countable structures, computably enumerable structures are one of the most important objects of investigation in computable model theory. Here we focus on the lattice structure of computably enumerable substructures of a given canonical computable structure. In particular, for a Turing degree d, we investigate the groups of d-computable automorphisms of the lattice of d-computably enumerable vector spaces, of the interval Boolean algebra Bη of the ordered set of rationals, and of the lattice of d-computably enumerable subalgebras of Bη. For these groups, we show that Turing reducibility can be used to substitute the group-theoretic embedding. We also prove that the Turing degree of the isomorphism types for these groups is the second Turing jump d′′ of d. PubDate: 2020-05-21

Abstract: Associative rings are considered. By a lattice isomorphism, or projection, of a ring R onto a ring Rφ we mean an isomorphism φ of the subring lattice L(R) of R onto the subring lattice L(Rφ) of Rφ. In this case Rφ is called the projective image of a ring R and R is called the projective preimage of a ring Rφ. Let R be a finite ring with identity and Rad R the Jacobson radical of R. A ring R is said to be local if the factor ring R/Rad R is a field. We study lattice isomorphisms of finite local rings. It is proved that the projective image of a finite local ring which is distinct from GF(\( {\mathrm{p}}^{{\mathrm{q}}^{\mathrm{n}}} \)) and has a nonprime residue field is a finite local ring. For the case where both R and Rφ are local rings, we examine interrelationships between the properties of the rings. PubDate: 2020-05-21

Abstract: We deal with questions concerning the completeness and stability of a class of injective acts and a class of weakly injective acts over a monoid S. The concepts of an injective S-act and of a weakly injective S-act are analogs of the concept of an injective module. In the theory of modules, the corresponding notions of injectivities in accordance with Baer’s criterion coincide. Also we will look into the completeness and stability of a class of principally weakly injective S-acts and a class of fg-weakly injective S-acts, which are analogs of p-injective modules and finitely injective modules. PubDate: 2020-05-21