 |
|
| RESIDUATED LATTICES: AN ALGEBRAIC GLIMPSE AT SUBSTRUCTURAL LOGICS, 151
| |
| | |
|
|
 To order this title, and for more information, click here
By
Nikolaos Galatos, School of Information Science, Japan Advanced Institute of Science and Technology
Peter Jipsen, Chapman University, Orange, USA
Tomasz Kowalski, Australian National University, Canberra, Australia
Hiroakira Ono, Japan Advanced Institute of Science and Technology, Ishikawa, Japan
Included in series
Studies in Logic and the Foundations of Mathematics,
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 negation is a classical one. We generalise it to the substructural
setting, identifying for each substructural logic its Glivenko equivalence class with smallest and largest element. This is also where
we begin investigating lattices of logics and varieties, rather than particular examples. We continue in this vein by presenting a number
of results concerning minimal varieties/maximal logics. A typical theorem there says that for some given well-known variety its subvariety
lattice has precisely such-and-such number of minimal members (where values for such-and-such include, but are not limited to, continuum,
countably many and two). In the last two chapters we focus on the lattice of varieties corresponding to logics without contraction. In
one we prove a negative result: that there are no 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 that finiteness properties, decidability and Glivenko
theorems are of clear interest to logicians, whereas semisimplicity and discriminator varieties are universal algebra par exellence.
It is for the reader to judge whether we succeeded in weaving these threads into a seamless fabric.
Audience
This book is intended for:
Research mathematicians and graduate students
and:
Computer scientists
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
| Bibliographic details |
Hardbound, 532 pages, publication date: APR-2007
ISBN-13: 978-0-444-52141-5
ISBN-10: 0-444-52141-0
Imprint: ELSEVIER
|
| Price and Ordering |
Price:
EUR 120
USD 132
GBP 83
|  |
Books and book related electronic products are priced in US dollars (USD), euro (EUR), and Great Britain Pounds (GBP). USD prices apply to the Americas and Asia Pacific. EUR prices apply in Europe and the Middle East. GBP prices apply to the UK and all other countries.
|
See also information about conditions of sale & ordering procedures, and links to our regional sales offices.
|
050/500
Last update: 27 Sep 2008
|
|
| |
| | |
|
|
| |
