Algebra and Logic最新文献

A multiring is a ring-like structure where the sum is multivalued and a hyperring is a multiring with a strong distributive property. With every multiring we associate a structural presheaf, and when that presheaf is a sheaf, we say that the multiring is geometric. A characterization of geometric von Neumann hyperrings is presented. And we build a von Neumann regular hull for multirings which is used in applications to algebraic theory of quadratic forms. Namely, we describe the functor Q, introduced by M. Marshall in [J. Pure Appl. Alg., 205, No. 2, 452-468 (2006)], as a left adjoint functor for the natural inclusion of the category of real reduced multirings (similar to real semigroups) into the category of preordered multirings and explore some of its properties. Next, we employ sheaf-theoretic methods to characterize real reduced hyperrings as certain geometric von Neumann regular real hyperrings and construct the functor V , a geometric von Neumann regular hull for a multiring. Finally, we look at some interesting logical and algebraic interactions between the functors Q and V that are useful for describing hyperrings in the image of the functor Q and that will allow us to explore the theory of quadratic forms for (formally) real semigroups.

The notion of a computable structure based on numberings with decidable equality is well established with a number of prominent results. Nevertheless, applied to strictly ordered fields, it fails to capture some natural properties and constructions for which decidability of equality is not assumed. For example, the field of primitive recursive real numbers is not computable, and there exists a computable real closed field with noncomputable maximal Archimedean subfields. We introduce the notion of an order positive field which aims to overcome these limitations. A general criterion is presented which decides when an Archimedean field is order positive. Using this criterion, we show that the field of primitive recursive real numbers is order positive and that the Archimedean parts of order positive real closed fields are order positive. We also state a program for further research.

We prove a theorem that generalizes Lemma 4 in the paper of A. S. Mamontov [Sib. Math. J., 54, No. 1, 114-118 (2013)] concerning the validity of the Baer–Suzuki theorem in groups of period 12. The results of the present work can be used in studying groups with a given set of element orders, also called a spectrum.

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.