Schur indices for reality-based algebras with two nonreal basis elements

This article discusses the representation theory of noncommutative algebras reality-based algebras with positive degree map over their field of definition. When the standard basis contains exactly two nonreal elements, the main result expresses the noncommutative simple component as a generalized quaternion algebra over its field of definition. The field of real numbers will always be a splitting field for this algebra, but there are noncommutative table algebras of dimension [Formula: see text] with rational field of definition for which it is a division algebra. The approach has other applications, one of which shows noncommutative association scheme of rank [Formula: see text] must have at least three symmetric relations.

Using the whole image and restrictions in the research of properties of some signature operations in Table Algebra

The features of the whole image relative to many binary relation, and restrictions on a binary relation on the set for some of the signature operations of Table Algebra are used in the work. Constructions of the whole image and restrictions are of general interest for Mathematics, and Table Algebra is a modern analogue of Codd's well-known Relational Algebra. It forms the theoretical foundation of modern query language databases. Elements of the carrier of Table Algebra specify relational table data structures, and signature operations are based on the basic table manipulations in Relational Algebra and SQL-like languages. The following results in the research of the features of the whole image were obtained: interconnections between the whole image and restrictions were found; the monotony and distribution of the whole image and restrictions on unions, a criterion of their emptiness and interconnections with first and second projection relations were proved; the whole image of the composition of relations and composition restrictions were found; the distribution of restriction on intersection of sets was set; the estimates of the distribution of the whole image of intersection and difference of sets were given; criteria for distribution of the whole image relative to the intersection and differences of sets were found. In addition, the clues were provided with the help of the whole image and restrictions on some of the signature operations of Table Algebra: intersection, union, difference, projection and joining. These representations allowed us to obtain some features of these operations, which derive directly from the features of the whole image and restrictions. It is supposed to get similar views on other signature operations of Table Algebras and to allocate their features arising from such representation. The obtained results can be used in the theory of Table Algebra as an approach to the research of the features of their signature operations, this can be used in query optimization in relational databases.