Residuated Lattices: An Algebraic Glimpse at Substructural Logics

The book is meant to serve two purposes. The first and the most obvious is the current state of the art in algebraic. The second, less important is algebraic logic. At the beginning, the second objective is predominant. Hence, in the first few chapters for logicians, a break course in nonclassical logics for algebraists, an introduction to residuated structures, an outline of proof theory - the celebrated Hauptsatz, or cut elimination theorem, among them. These results are a discussion of interconnections between logic and algebra, where we have the same coin. We envisage the initial chapters could be used as a textbook for a graduate course, possibly 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 say. These include Dedekind-McNeille completions and canonical extensions. Completions are already in the finite model of property, generation of varieties of finite members, and finite embeddability. The algebraic analysis of the cut elimination that follows, also takes recourse to completions. Decidability of logics, equational and quasi-equational theories comes next, where we show theoretical methods, like semantic tools like Rabin's theorem work better for big ones. Then we turn to Glivenko's theorem, which says it is an intuitionistic tautology. We generalise it to the substructural setting, identifying for each of them the Glivenko equivalence class with smallest and largest element. This is also where we are investigating lattices of logics and varieties, rather than particular examples. We have a minimal varieties / maximal logics. A typical theorem there says that for some given well-known variety its sub-lattice has the exact number of minimal members (continuum, countably many and two) ). In the last two chapters in focus on the lattice of varieties. In one we prove a negative result: that there are nontrivial splittings in that variety. In the other, we prove a positive one: that semisimple varieties coincide with discriminator ones.

Within the second, more technical part of the book. Another transition process may be traced. Namely, we begin with logically inclined technicalities and end with algebraically inclined ones. Here, perhaps, algebraic rendering of Glivenko theorems marks the equilibrium point, at least in the sense of finiteness properties, decidability. It is for the reader to judge whether we are able to do it.

- Considers both the algebraic and logical perspective within a common framework.
- Written by experts in the area.
- Easily 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 proposed research directions.
- Provides a concise introduction to the subject and leads directly to research topics.
- The ideas from algebra and logic are developed in a hand-in-hand and the connections are shown in every level.

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