Algebra and Logic最新文献

We prove that for every natural number n, there exists a natural number N (n) such that every multilinear skew-symmetric polynomial in N (n) or more variables which vanishes in the free associative algebra also vanishes in any n-generated alternative algebra over a field of characteristic 0. Previously, a similar result was proved only for a series of skew-symmetric polynomials constructed by I. P. Shestakov in [Algebra and Logic, 16, No. 2, 153-166 (1977)].

An axiomatizability criterion is found for the class of subdirectly irreducible S-acts over a commutative monoid. As a corollary, a number of properties are presented which a commutative monoid should satisfy provided that the class of subdirectly irreducible acts over it is axiomatizable. The question about a complete description of monoids over which the class of subdirectly irreducible acts is axiomatizable remains open even for the case of a commutative monoid.

The notion of an exponential R-group, where R is an arbitrary associative ring with unity, was introduced by R. Lyndon. Myasnikov and Remeslennikov refined the notion of an R-group by introducing an additional axiom. In particular, the new concept of an exponential MR-group (R-ring) is a direct generalization of the concept of an R-module to the case of noncommutative groups. We come up with the notions of a variety of MR-groups and of tensor completions of groups in varieties. Abelian varieties of MR-groups are described, and various definitions of nilpotency in this category are compared. It turns out that the completion of a 2-step nilpotent MR-group is 2-step nilpotent.

Inference rules are examined which are admissible immediately in all residually finite extensions of S4 possessing the weak cocover property. An explicit basis is found for such WCP-globally admissible rules. In case of tabular logics, the basis is finite, and for residually finite extensions, the independency of an explicit basis is proved.

For a 5-dimensional 2-step Carnot group G_3,2 with a codimension 2 horizontal distribution, we prove that any two points u, v ∈ G_3,2 can be joined on it by a horizontal broken line consisting of at most three segments. A multi-dimensional generalization of this result is given.

An S-ring (Schur ring) is said to be central if it is contained in the center of a group ring. We introduce the notion of a generalized Schur group, i.e., a finite group such that all central S-rings over this group are Schurian. It generalizes the notion of a Schur group in a natural way, and for Abelian groups, the two notions are equivalent. We prove basic properties and present infinite families of non-Abelian generalized Schur groups.

We prove that if (mathcal{A}) = (A,⋅) is a group computable in polynomial time (P-computable), then there exists a P-computable group (mathcal{B}) = (B,∙) ≅ (mathcal{A},) in which the operation x⁻¹ is also P-computable. On the other hand, we show that if the center (Zleft(mathcal{A}right)) of a group A contains an element of infinite order, then under some additional assumptions, there exists a P-computable group ({mathcal{B}}{prime}=left({B}{prime},cdot right)cong mathcal{A}) in which the operation x⁻¹ is not primitive recursive. Also the following general fact in the theory of P-computable structures is stated: if (mathcal{A}) is a P-computable structure and E ⊆ A² is a P-computable congruence on (mathcal{A},) then the quotient structure (mathcal{A}/E) is isomorphic to a P-computable structure.