# Residuated Lattices: An Algebraic Glimpse at Substructural Logics, Volume 151

## 1st Edition

**Authors:**Nikolaos Galatos Peter Jipsen Tomasz Kowalski Hiroakira Ono

**Print ISBN:**9780444550668

**eBook ISBN:**9780080489643

**Imprint:**Elsevier Science

**Published Date:**25th April 2007

**Page Count:**532

## Description

The book is meant to serve two purposes. The first and more obvious one is to present state of the art results in algebraic research into residuated structures related to substructural logics. The second, less obvious but equally important, is to provide a reasonably gentle introduction to algebraic logic. At the beginning, the second objective is predominant. Thus, in the first few chapters the reader will find a primer of universal algebra for logicians, a crash course in nonclassical logics for algebraists, an introduction to residuated structures, an outline of Gentzen-style calculi as well as some titbits of proof theory - the celebrated Hauptsatz, or cut elimination theorem, among them. These lead naturally to a discussion of interconnections between logic and algebra, where we try to demonstrate how they form two sides of the same coin. We envisage that the initial chapters could be used as a textbook for a graduate course, perhaps entitled Algebra and Substructural Logics. As the book progresses the first objective gains predominance over the second. Although the precise point of equilibrium would be difficult to specify, it is safe to say that we enter the technical part with the discussion of various completions of residuated structures. These include Dedekind-McNeille completions and canonical extensions. Completions are used later in investigating several finiteness properties such as the finite model property, generation of varieties by their finite members, and finite embeddability. The algebraic analysis of cut elimination that follows, also takes recourse to completions. Decidability of logics, equational and quasi-equational theories comes next, where we show how proof theoretical methods like cut elimination are preferable for small logics/theories, but semantic tools like Rabin's theorem work better for big ones. Then we turn to Glivenko's theorem, which says that a formula is an intuitionistic tautology if and only if its double neg

## Key Features

- Considers both the algebraic and logical perspective within a common framework.
- Written by experts in the area.
- Easily accessible to graduate students and researchers from other fields.
- Results summarized in tables and diagrams to provide an overview of the area.
- Useful as a textbook for a course in algebraic logic, with exercises and suggested research directions.
- Provides a concise introduction to the subject and leads directly to research topics.
- The ideas from algebra and logic are developed hand-in-hand and the connections are shown in every level.

## Readership

This book is intended for: Research mathematicians and graduate students and: Computer scientists

## Table of Contents

Contents

List of Figures

List of Tables

Introduction

Chapter 1. Getting started

Chapter 2. Substructural logics and residuated lattices

Chapter 3. Residuation and structure theory

Chapter 4. Decidability

Chapter 5. Logical and algebraic properties

Chapter 6. completions and finite embeddability

Chapter 7. Algebraic aspects of cut elimination

Chapter 8. Glivenko theorems

Chapter 9. Lattices of logics and varieties

Chapter 10. Splittings

Chapter 11. Semisimplicity

Bibliography

Index

## Details

- No. of pages:
- 532

- Language:
- English

- Copyright:
- © Elsevier Science 2007

- Published:
- 25th April 2007

- Imprint:
- Elsevier Science

- eBook ISBN:
- 9780080489643

- Hardcover ISBN:
- 9780444521415

- Paperback ISBN:
- 9780444550668

## About the Author

### Nikolaos Galatos

### Affiliations and Expertise

School of Information Science, Japan Advanced Institute of Science and Technology

### Peter Jipsen

### Affiliations and Expertise

Chapman University, Orange, USA

### Tomasz Kowalski

### Affiliations and Expertise

Australian National University, Canberra, Australia

### Hiroakira Ono

### Affiliations and Expertise

Japan Advanced Institute of Science and Technology, Ishikawa, Japan

## Reviews

The book is meant to serve two purposes. The first and more obvious one is to present state of the art results in algebraic research into residuated structures related to substructural logics. The second, less obvious but equally important, is to provide a reasonably gentle introduction to algebraic logic. At the beginning, the second objective is predominant. Thus, in the first few chapters the reader will find a primer of universal algebra for logicians, a crash course in nonclassical logics for algebraists, an introduction to residuated structures, an outline of Gentzen-style calculi as well as some titbits of proof theory - the celebrated Hauptsatz, or cut elimination theorem, among them. These lead naturally to a discussion of interconnections between logic and algebra, where we try to demonstrate how they form two sides of the same coin. We envisage that the initial chapters could be used as a textbook for a graduate course, perhaps entitled Algebra and Substructural Logics. As the book progresses the first objective gains predominance over the second. Although the precise point of equilibrium would be difficult to specify, it is safe to say that we enter the technical part with the discussion of various completions of residuated structures. These include Dedekind-McNeille completions and canonical extensions. Completions are used later in investigating several finiteness properties such as the finite model property, generation of varieties by their finite members, and finite embeddability. The algebraic analysis of cut elimination that follows, also takes recourse to completions. Decidability of logics, equational and quasi-equational theories comes next, where we show how proof theoretical methods like cut elimination are preferable for small logics/theories, but semantic tools like Rabin's theorem work better for big ones. Then we turn to Glivenko's theorem, which says that a formula is an intuitionistic tautology if and only if its double neg