Details

Subject(s)

• Algebra, Universal [Browse]
• Algebraic logic [Browse]

Author

• Paoli, Francesco [Browse]
• Tsinakis, Constantine [Browse]

Series

• Mathematical surveys and monographs ; Volume 277. [More in this series]
• Mathematical Surveys and Monographs ; Volume 277

Summary note

This book is an introduction to residuated structures, viewed as a common thread binding together algebra and logic. The framework includes well-studied structures from classical abstract algebra such as lattice-ordered groups and ideals of rings, as well as structures serving as algebraic semantics for substructural and other non-classical logics. Crucially, classes of these structures are studied both algebraically, yielding a rich structure theory along the lines of Conrad's program for lattice-ordered groups, and algorithmically, via analytic sequent or hypersequent calculi. These perspectives are related using a natural notion of equivalence for consequence relations that provides a bridge offering benefits to both sides. Algorithmic methods are used to establish properties like decidability, amalgamation, and generation by subclasses, while new insights into logical systems are obtained by studying associated classes of structures.The book is designed to serve the purposes of novices and experts alike. The first three chapters provide a gentle introduction to the subject, while subsequent chapters provide a state-of-the-art account of recent developments in the field.

Bibliographic references

Includes bibliographical references and index.

Source of description

Description based on print version record.

Contents

• Intro
• Blank Page
• Contents
• Introduction
• Overview of the book
• Chapter 1. Order and residuation
• 1.1. Partially ordered sets and lattices
• 1.2. Residuated maps
• 1.3. Closure and co-closure operators
• 1.4. Residuated lattices: the algebras of logic
• 1.5. Nuclei and co-nuclei
• 1.6. Historical excursus
• 1.7. Bibliographical remarks
• Chapter 2. Proof systems
• 2.1. Rules and derivations
• 2.2. A proof system for lattices
• 2.3. The full Lambek calculus
• 2.4. Adding structural rules
• 2.5. Hypersequent calculi
• 2.6. Historical excursus
• 2.7. Bibliographical remarks
• Chapter 3. Consequence relations
• 3.1. Abstract consequence relations
• 3.2. Equational consequence relations
• 3.3. Equivalence of consequence relations
• 3.4. Residuated lattices and the full Lambek calculus
• 3.5. Historical excursus
• 3.6. Bibliographical remarks
• Chapter 4. Structure theory
• 4.1. Convex subuniverses
• 4.2. Polars and prime convex subuniverses
• 4.3. Congruence relations
• 4.4. Normal convex subuniverse generation
• 4.5. Bibliographical remarks
• Chapter 5. Semilinearity and distributivity
• 5.1. Equational bases for semilinear varieties
• 5.2. Densifiable varieties
• 5.3. Representations of distributive varieties
• 5.4. Generation and decidability results
• 5.5. Bibliographical remarks
• Chapter 6. Cancellativity
• 6.1. Cancellative residuated lattices
• 6.2. Lattice-ordered groups of left quotients
• 6.3. A categorical equivalence
• 6.4. Bibliographical remarks
• Chapter 7. Divisibility
• 7.1. GBL-algebras and GMV-algebras
• 7.2. Direct decomposition
• 7.3. Ordinal decomposition
• 7.4. Cone algebras and negative cones
• 7.5. A categorical equivalence
• 7.6. Strongly simple GBL-algebras
• 7.7. Bibliographical remarks
• Chapter 8. Bridges between algebra and logic.
• 8.1. The amalgamation property
• 8.2. The congruence extension property
• 8.3. Interpolation properties
• 8.4. Amalgamation in varieties of residuated lattices
• 8.5. Bibliographical remarks
• Chapter 9. Finite embeddings and finite models
• 9.1. The finite embeddability property
• 9.2. Finite model properties
• 9.3. Join-extensions and join-completions
• 9.4. The FEP for varieties of residuated lattices
• 9.5. Bibliographical remarks
• Appendix A. Open problems
• Structure theory
• Proof systems
• Amalgamation and interpolation
• Decidability
• First-order and modal substructural logics
• Appendix B. Basic notions of universal algebra
• Algebras and subalgebras
• Homomorphisms and congruences
• Direct and subdirect products
• Varieties and free algebras
• Equational classes and the HSP theorem
• Ultraproducts
• Index
• Bibliography.

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ISBN

• 9781470475512 ((ebook))
• 1-4704-7551-0

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