By
R. Hirsch, University College, London, UK
I. Hodkinson, Imperial College, London, UK
Description
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.
Included in series
Studies in Logic and the Foundations of Mathematics
