Bibliothèque de L'IMIST/CNRST

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. After an introduction describing the authors' perspective on the material, the text proper has six parts. The lengthy first part is devoted to background material, including the formal definitions of relation algebras, cylindric algebras, their basic properties, and some connections between them. Examples are given. Part 1 ends with a short survey of other work beyond the scope of the book. In part 2, games are introduced, and used to axiomatise various classes of algebras. Part 3 discusses approximations to representability, using bases, relation algebra reducts, and relativised representations. Part 4 presents some constructions of relation algebras, including Monk algebras and the 'rainbow construction', and uses them to show that various classes of representable algebras are non-finitely axiomatisable or even non-elementary. Part 5 shows that the representability problem for finite relation algebras is undecidable, and then in contrast proves some finite base property results. Part 6 contains a condensed summary of the book, and a list of problems. There are more than 400 exercises. The book is generally self-contained on relation algebras and on games, and introductory text is scattered throughout. Some familiarity with elementary aspects of first-order logic and set theory is assumed, though many of the definitions are given. Chapter 2 introduces the necessary universal algebra and model theory, and more specific model-theoretic ideas are explained as they arise.

Includes bibliographical references (p. 629-654) and indexes.

Description based on print version record.

Cover -- Contents -- Preface -- Foreword -- Chapter 1. Introduction -- 1.1 History -- 1.2 To the games -- 1.3 Non-finite axiomatisability -- 1.4 Approximations to representability -- 1.5 Constructions of algebras -- 1.6 Some remarks on methods -- 1.7 Summary of contents -- Part I: Algebras of Relations -- Chapter 2. Preliminaries -- 2.1 Foundations -- 2.2 Model theory -- 2.3 Boolean algebras -- 2.4 Products and ultraproducts -- 2.5 Boolean algebras with operators -- 2.6 Varieties and quasi-varieties of BAOs -- 2.7 Aspects of duality for BAOs -- Chapter 3. Binary relations and relation algebra -- 3.1 Algebraic logic -- 3.2 Binary relations -- 3.3 Relation algebras -- 3.4 Representations of relation algebras -- Chapter 4. Examples of relation algebras -- 4.1 Set algebras -- 4.2 Group relation algebras -- 4.3 n-variable logic -- 4.4 Examples -- 4.5 The Lyndon algebras -- Chapter 5. Relativisation and cylindric algebras -- 5.1 Relativisation -- 5.2 Weakly representable relation algebras -- 5.3 Cylindric algebras -- 5.4 Substitutions in cylindric algebras -- 5.5 Relativised cylindric algebras -- 5.6 Relation algebra reducts of cylindric algebras -- 5.7 Relation algebra reducts of other cylindric-type algebras -- Chapter 6. Other approaches to algebras of relations -- 6.1 Diagonal-free algebras -- 6.2 Polyadic algebra -- 6.3 Pinter's substitution algebras -- 6.4 Finitisation problem -- 6.5 Decidability -- 6.6 Amalgamation -- 6.7 Technical innovations -- 6.8 Applications -- Part II: Games -- Chapter 7. Games and networks -- 7.1 Networks -- 7.2 Refining networks -- 7.3 All weakly associative algebras have relativised representations -- 7.4 Games on relation algebra networks -- 7.5 Strategies -- 7.6 Games and representations of relation algebras -- 7.7 Networks for cylindric algebras -- 7.8 Games for cylindric algebra networks -- 7.9 Games for temporal constraint handling -- 7.10 Summary of chapter -- Chapter 8. Axiomatising representable relation algebras and cylindric algebras -- 8.1 The relation algebra case -- 8.2 An axiomatisation using 'Q-operators' -- 8.3 Axiomatising RCAd for 3 <= d <? -- 8.4 Axiomatising RCA a for infinite a -- Chapter 9. Axiomatising pseudo-elementary classes -- 9.1 Introduction -- 9.2 Pseudo-elementary classes -- 9.3 Examples -- 9.4 Model theory of pseudo-elementary classes -- 9.5 More explicit axioms -- 9.6 Axiomatising pseudo-elementary classes -- 9.7 Generalised Q-operators -- Chapter 10. Game trees -- 10.1 Trees, and games on them -- 10.2 Strategies -- 10.3 Examples -- 10.4 Formulas expressing a winning strategy -- 10.5 Games and non-finite axiomatisability -- Chapter 11. Atomic networks -- 11.1 Introduction -- 11.2 Atomic networks and games -- 11.3 Alternative views of the game -- 11.4 Atomic games and complete representations -- 11.5 Axio.