E-Book Overview
Relation algebras are algebras arising from the study of binary relations. They form a part of the field of algebraic logic, and have applications in proof theory, modal logic, and computer science. This research text uses combinatorial games to study the fundamental notion of representations of relation algebras. Games allow an intuitive and appealing approach to the subject, and permit substantial advances to be made. The book contains many new results and proofs not published elsewhere. It should be invaluable to graduate students and researchers interested in relation algebras and games.
E-Book Content
Preface Relation algebras are algebras arising from the study of binary relations. They form a part of the field of algebraic logic, and have applications in proof theory, modal logic, and computer science. This book uses combinatorial games to develop some of the theory of relation algebras, focusing on the fundamental notion of representation. Games allow an intuitive and appealing approach to the subject, and permit substantial advances to be made. The introduction explains our perspective on the material. We hope that the book will be used by graduate students and researchers interested in relation algebras and games. The book proper is divided into six parts. The lengthy first part presents some necessary background material, including the formal definitions of relation algebras, cylindric algebras, their basic properties, and some connections between them. Examples are given. Part I ends with a short survey of other work beyond the scope of the book. In part II we introduce the games, and use them to axiomatise various classes of algebras. Part III discusses approximations to representability, using relational bases, hyperbases, relation algebra reducts, and relativised representations. In part IV we present some constructions of relation algebras, including Monk al