Relation Algebras by Games 1st Edition by R Hirsch, I Hodkinson ISBN 0444509321 9780444509321

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ISBN 10: 0444509321
ISBN 13: 9780444509321
Author: R Hirsch, I Hodkinson

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.

Relation Algebras by Games 1st Table of contents:

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 < ω
8.4 Axiomatising RCA α for infinite α
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 Axioms for complete representability?
Part III: Approximations
Chapter 12. Relational, cylindric, and hyperbases
12.1 Hypernetworks
12.2 Relational bases and hyperbases
12.3 Elementary properties of bases
12.4 Games
12.5 The variety RAn
12.6 Maddux’s bases
12.7 Cylindric bases and homogeneous representations
Chapter 13. Approximations to RRA
13.1 Representation theory
13.2 From relativised representations to relation algebra reducts
13.3 From reducts to relational bases
13.4 From reducts to hyperbases
13.5 From bases to relativised representations
13.6 From smooth to hyperbasis
13.7 Summary and discussion
13.8 Equational axioms for RAn and SRaCAn
Part IV: Constructing Relation Algebras
Chapter 14. Strongly representable relation algebra atom structures
14.1 Introduction
14.2 SRAS is not an elementary class
14.3 Consequences of the theorem
14.4 Maddux’s construction
Chapter 15. Non-finite axiomatisability of SRaCAn+1 over SRaCAn
15.1 Outline of chapter
15.2 The algebras u(n,r) and Cr
15.3 u(n,r) ε SRaCAn
15.4 u(n,r) ε/ SRaCAn+1
15.5 3 can win Gr m,n+1 (u(n, r), Λ)
15.6 Non-finite axiomatisability
15.7 Proof theory
Chapter 16. The rainbow construction for relation algebras
16.1 Ehrenfeucht-Fraissé ‘forth’ games
16.2 The rainbow algebra AA,B
16.3 How V can win G(AA,8)
16.4 How 3 can win G(AA,8)
16.5 Modifications to the rainbow algebra
Chapter 17. Applying the rainbow construction
17.1 Non-finite axiomatisability of RRA
17.2 Complete representations
17.3 There is no n-variable equational axiomatisation of RRA
17.4 RAn+l is not finitely based over RAn
17.5 Infinite-dimensional bases and relativised representations
17.6 Weakly representable relation algebras
17.7 Completions
Part V: Decidability
Chapter 18. Undecidability of the representation problem for finite algebras
18.1 Introduction
18.2 The tiling problem
18.3 The definition of RA(τ)
18.4 Games
18.5 Winning 3-strategy implies tiling
18.6 RA(τ) ε SRtCA5 implies tiling
18.7 Tiling implies winning 3-strategy
18.8 Conclusion
18.9 Weak representability is undecidable
18.10 Undecidability of equational theories
Chapter 19. Finite base property
19.1 Introduction
19.2 Guarded fragments
19.3 The finite base property
19.4 Finite base property for WA
19.5 Finite algebra on finite base property for RAn
19.6 The finite algebra on finite base property for SRaCAn?
Part VI: Epilogue
Chapter 20. Brief summary
20.1 Basic definitions
20.2 Games for representability
20.3 Relativised representations, bases, reducts
20.4 The rainbow construction
20.5 Atom structures
20.6 Decidability
20.7 Summary of relations between the classes
20.8 Summary of properties of classes
Chapter 21. Problems
Bibliography
Symbol index
Subject index

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