Table Algebra of Infinite Tables, Multiset Table Algebra, and their Relationship

The paper is focused on some theoretical questions of the table databases. Two mathematical formalisms such as table algebra of infinite tables and multiset table algebra are considered. Basic definitions referring to these formalisms are given. This paper also addresses the issue of the relationship between table algebra of infinite tables and multiset table algebra. It is proved that table algebra of infinite tables is not a subalgebra of multiset table algebra since it is not closed in relation to some signature operations of multiset table algebra. These signature operations are determined.

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.

The interrelations between intersection, union and other signature operations in table algebra

Currently, databases are widely used in almost all areas of human activity. For all variety of different types of databases the most common are relational (table) databases, mathematical model of which was proposed by E. Codd. From mathematical point of view, a relational database is a finite set of finite relations between different predefined sets of basic data. Table algebra introduced by V.N. Red’ko and D.B. Buy is based on Codd’s relational algebra and significantly improves it. It formed the theoretical foundation of modern database query language. Elements of the carrier of table algebra specify relational data structures, and signature operations are based on the basic table manipulations in relational algebra and SQL-like languages. One of the most actual tasks in relational and table algebras is the problem of equivalent transformation of expressions in order to minimize or reduce them to a standard form; it is one of the stages of query optimization, and can also significantly reduce the processing time of information in relational database management systems. For the decision of this problem the interrelations between the basic table operations are used. In the present, a significant number of such interrelations have been established, most of which for the general case are performed as inclusions. The author has found criteria for the transition of some such inclusions into equalities. These criteria are expressed in terms of the active domains of the tables and are natural. In this paper, the interrelations of the intersection and the union of tables with other signature operations of table algebras: difference, selection, projection, saturation, active complement, join, renaming of attributes are considered.

Outer Set Operations of Table Algebra of Infinite Tables

Table algebra of infinite tables is considered. The signature of table algebra of infinite tables is filled up with outer set operations. A formal mathematical semantics of these operations is defined.

On the Multiplicities of Characters in Table Algebras

In this paper we show that every module of a table algebra can be considered as a faithful module of some quotient table algebra. Also we prove that every faithful module of a table algebra determines a closed subset that is a cyclic group. As a main result we give some information about multiplicities of characters in table algebras.

TABLE ALGEBRAS OF RANK 3 AND ITS APPLICATIONS TO STRONGLY REGULAR GRAPHS

A table algebra is called quasi self-dual if there exists a permutation on the set of primitive idempotents under which any Krein parameter is equal to its corresponding structure constants. In this paper we investigate the question of when a table algebra of rank 3 is quasi self-dual. As a direct consequence we find necessary and sufficient conditions for the Bose–Mesner algebra of a given strongly regular graph to be quasi self-dual. In fact, our result generalizes the well-known Delsarte's characterization of a self-duality of the Bose–Mesner algebra of a strongly regular graph given in [P. Delsarte, An algebraic approach to the association schemes of coding theory, Philips Res. Rep. Suppl.10 (1973) 1–97]. Among our results we determine conditions under which the Krein parameters of an integral table algebra of rank 3 are non-negative rational numbers.