Universal Algebra, Algebraic Logic, and Databases

Modern algebra, which not long ago seemed to be a science divorced from real life, now has numerous applications. Many fine algebraic structures are endowed with meaningful contents. Now and then practice suggests new and unexpected structures enriching algebra. This does not mean that algebra has become merely a tool for applications. Quite the contrary, it significantly benefits from the new connections. The present book is devoted to some algebraic aspects of the theory of databases. It consists of three parts. The first part contains information about universal algebra, algebraic logic is the subject of the second part, and the third one deals with databases. The algebraic material of the flI'St two parts serves the common purpose of applying algebra to databases. The book is intended for use by mathematicians, and mainly by algebraists, who realize the necessity to unite theory and practice. It is also addressed to programmers, engineers and all potential users of mathematics who want to construct their models with the help of algebra and logic. Nowadays, the majority of professional mathematicians work in close cooperation with representatives of applied sciences and even industrial technology. It is neces sary to develop an ability to see mathematics in different particular situations. One of the tasks of this book is to promote the acquisition of such skills.

Klappentext
This volume is devoted to the development of an algebraic model of databases. The first chapter presents a general introduction. The following sixteen chapters are divided into three main parts. Part I deals with various aspects of universal algebra. The chapters of Part I discuss topics such as sets, algebras and models, fundamental structures, categories, the category of sets, topoi, fuzzy sets, varieties of algebras, axiomatic classes, category algebra and algebraic theories.Part II deals with different approaches to the algebraization of predicate calculus. This material is intended to be applied chiefly to databases, although some discussion of pure algebraic applications is also given. Discussed here are topics such as Boolean algebras and propositional calculus, Halmos algebras and predicate calculus, connections with model theory, and the categorial approach to algebraic logic.Part III is concerned specifically with the algebraic model of databases, which considers the database as an algebraic structure. Topics dealt with in this part are the algebraic aspects of databases, their equivalence and restructuring, symmetries and the Galois theory of databases, and constructions in database theory. The volume closes with a discussion and conclusions, and an extensive bibliography. For mathematicians, computer scientists and database engineers, with an interest in applications of algebra and logic.

Inhalt
0 General View on Objectives and Contents of The Book.- I Universal Algebra.- 1 Sets, Algebras, Models.- 2 Fundamental Structures.- 3 Categories.- 4 The Category of Sets. Topoi. Fuzzy Sets.- 5 Varieties of Algebras. Axiomatizable Classes.- 6 Category Algebra and Algebraic Theories.- II Algebraic Logic.- 7 Boolean Algebras and Propositional Calculus.- 8 Halmos Algebras and Predicate Calculus.- 9 Specialized Halmos Algebras.- 10 Connections With Model Theory.- 11 The Categorial Approach to Algebraic Logic.- III Databases Algebraic Aspects.- 12 Algebraic Model of a Database.- 13 Equivalence and Reorganization of Databases.- 14 Symmetries of Relations and Galois Theory of Databases.- 15 Constructions in Database Theory.- 16 Discussion and Conclusion.