Residuated structures in algebra and logic
 Responsibility
 George Metcalfe, Francesco Paoli, Constantine Tsinakis.
 Publication
 Providence, Rhode Island : American Mathematical Society, [2023]
 Copyright notice
 ©2023
 Physical description
 1 online resource (xiii, 265 pages) : illustrations.
 Series
 Mathematical surveys and monographs ; no. 277.
Online
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Description
Creators/Contributors
 Author/Creator
 Metcalfe, George, author.
 Contributor
 Paoli, Francesco, author.
 Tsinakis, Constantine, author.
Contents/Summary
 Bibliography
 Includes bibliographical references (pages 253265) and index.
 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 coclosure operators
 1.4. Residuated lattices: the algebras of logic
 1.5. Nuclei and conuclei
 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. Latticeordered groups of left quotients
 6.3. A categorical equivalence
 6.4. Bibliographical remarks
 Chapter 7. Divisibility
 7.1. GBLalgebras and GMValgebras
 7.2. Direct decomposition
 7.3. Ordinal decomposition
 7.4. Cone algebras and negative cones
 7.5. A categorical equivalence
 7.6. Strongly simple GBLalgebras
 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. Joinextensions and joincompletions
 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
 Firstorder 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
 Summary
 An introduction to residuated structures, a common thread binding together algebra and logic, 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 stateoftheart account of recent developments in the field.
Subjects
 Subjects
 Algebraic logic.
 Algebra, Universal.
 Logique algébrique.
 Algèbre universelle.
 Algebra, Universal
 Algebraic logic
 Mathematical logic and foundations  General logic  Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics).
 Mathematical logic and foundations  Model theory  Equational classes, universal algebra.
 Mathematical logic and foundations  Proof theory and constructive mathematics  Linear logic and other substructural logics.
 Mathematical logic and foundations  Algebraic logic  Lattices and related structures.
 Order, lattices, ordered algebraic structures  Distributive lattices  MValgebras.
 Order, lattices, ordered algebraic structures  Ordered structures  Ordered semigroups and monoids.
 Order, lattices, ordered algebraic structures  Ordered structures  Ordered groups.
Bibliographic information
 Publication date
 2023
 Copyright date
 2023
 Series
 Mathematical surveys and monographs ; volume 277
 ISBN
 1470475510 (electronic bk.)
 9781470475512 (electronic bk.)
 9781470469856
 1470469855