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

Abstract

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.