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.
General
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