- Print ISBN 9780444509321
- Electronic ISBN 9780080540450
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
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
I Algebras of Relations.
2.2 Model theory.
2.2.2 Semantics - structures.
2.2.3 Models, validity.
2.2.4 Homomorphisms, embeddings, substructures.
2.2.5 Generating sets.
2.2.6 Compactness, Lowenheim-Skolem-Tarski theorems.
2.2.7 Relativisation, interpretations, second-order logic.
2.3 Boolean algebras.
2.3.1 Definition and examples.
2.3.3 Dense sets.
2.3.4 Ideals, filters, ultrafilters.
2.3.5 Representations of boolean algebras.
2.3.6 Canonical extensions.
2.3.7 Infinite sums and products.
2.3.8 Complete representations.
2.3.9 Completions of boolean algebras.
2.4 Products and ultraproducts.
2.4.2 Ultraproducts, ultrapowers.
2.5 Boolean algebras with operators.
2.5.2 Homomorphisms and ideals.
2.5.3 Completely additive and conjugated algebras.
2.5.4 Completions of BAOs.
2.6 Varieties and quasi-varieties of BAOs.
2.6.1 Basic concepts.
2.6.2 HSP notation and Birkhoff's theorem.
2.6.3 Subdirect products.
2.6.4 Discriminator varieties.