Universal Algebra, Algebraic Logic, and Databases - B. Plotkin

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Résumé :
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. br/ 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. br/ 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. br/ For mathematicians, computer scientists and database engineers, with an interest in applications of algebra and logic. br/

Sommaire:
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.