# Relation Algebras by Games

Gulf Professional Publishing, Aug 15, 2002 - Mathematics - 710 pages
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.

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### Contents

 Chapter 1 1 Introduction Part 23 I Algebras of Relations Part 213 II Games Part 353 III Approximations Part 439 IV Constructing Relation Algebras Part 537 V Decidability
 Part VI Epilogue 607 Bibliography 629 Symbol index 655 Subject index 667 Copyright

### About the author (2002)

Robin Hirsch is truly a Renaissance man. Writer, performer, producer, restaurateur, former Oxford and Fulbright scholar, he is the author of the acclaimed memoir, Last Dance at the Hotel Kempinski, and of Mosaic: Fragments of a Jewish Life, his award-winning solo performance cycle with which he has toured across the Unites States and Europe.
When he is not writing or performing, the author can usually be found at the Cornelia Street Cafe in Greenwich Village, of which he is a co-owner. On its tenth anniversary (in 1987), Mayor Koch proclaimed it "a cultural as well as a culinary landmark." (it also serves the best french fries his editor has ever tasted.)

Ha is the pen (or should we say "paintbrush?") name of an acclaimed artist whose illustrations have appeared in major magazines both in the United States and abroad, including a number of covers for the New Yorker. His work has won numerous awards and has been exhibited internationally. This is his first children's book. A native of Canada, Ha lives in Los Angeles.