Synopses & Reviews
The congruences of a lattice form the congruence lattice. In the past half-century, the study of congruence lattices has become a large and important field with a great number of interesting and deep results and many open problems. This self-contained exposition by one of the leading experts in lattice theory, George Grätzer, presents the major results on congruence lattices of finite lattices featuring the author's signature "Proof-by-Picture" method and its conversion to transparencies. Key features: * Includes the latest findings from a pioneering researcher in the field * Insightful discussion of techniques to construct "nice" finite lattices with given congruence lattices and "nice" congruence-preserving extensions * Contains complete proofs, an extensive bibliography and index, and nearly 80 open problems * Additional information provided by the author online at: http://www.maths.umanitoba.ca/homepages/gratzer.html/ The book is appropriate for a one-semester graduate course in lattice theory, yet is also designed as a practical reference for researchers studying lattices.
Review
"The book is self-contained, with many detailed proofs presented that can be followed step-by-step. [I]n addition to giving the full formal detals of the proofs, the author chooses a somehow more pedagogical way that he calls Proof-by-Picture, somehow related to the combinatorial (as opposed to algebraic) nature of many of the presented results. I believe that this book is a much-needed tool for any mathematician wishing a gentle introduction to the field of congruences representations of finite lattices, with emphasisi on the more 'geometric' aspects." --Mathematical Reviews "...covers very many interesting results on congruence lattices of finite distributive lattices and includes a wealth of open problems. The author's decision to provide a graphical sketch of proofs before formal proofs works very well. The book is very well suited for self-study as an entrée to this research area for readers already acquainted with the basics of lattice theory." --SIGACT NEWS "There exist a lot of interesting results in this area of lattice theory, and some of them are presented in this book. The monograph under review is an exceptional work in lattice theory, like all the contributions by this author. ... The way this book is written makes it extremely interesting for the specialists in the field but also for the students in lattice theory. Moreover, the author provides a series of companion lectures which help the reader to approach the Proof-by-Picture sections." (Cosmin Pelea, Studia Universitatis Babes-Bolyai Mathematica, Vol. LII (1), 2007)
Synopsis
The congruences of a lattice form the congruence lattice. In the past half-century, the study of congruence lattices has become a large and important field with a great number of interesting and deep results and many open problems. This self-contained exposition by one of the leading experts in lattice theory, George Gr?tzer, presents the major results on congruence lattices of finite lattices featuring the author's signature Proof-by-Picture" method and its conversion to transparencies.
Key features:
- Includes the latest findings from a pioneering researcher in the field
- Insightful discussion of techniques to construct "nice" finite lattices with given congruence lattices and "nice" congruence-preserving extensions
- Contains complete proofs, an extensive bibliography and index, and nearly 80 open problems
- Additional information provided by the author online at: www.maths.umanitoba.ca/homepages/gratzer.html/
The book is appropriate for a one-semester graduate course in lattice theory, yet is also designed as a practical reference for researchers studying lattices."
Synopsis
Self-contained exposition presents the major results on congruence lattices of finite lattices Includes the latest findings from a pioneering researcher in the field Features the author's signature "Proof-by-Picture" method and its conversion to transparencies Contains complete proofs, an extensive bibliography and index, and nearly 80 open problems Excellent grad text and reference
Table of Contents
* Table of Notation * Picture Gallery * Preface and Acknowledgment * Introduction Part I: A Brief Introduction to Lattices * Basic Concepts * Special Concepts * Congruences Part II: Basic Techniques * Chopped Lattices * Boolean Triples * Cubic Extensions Part III: Representation Theorems * The Dilworth Theorem * Minimal Representations * Semimodular Lattices * Modular Lattices * Uniform Lattices Part IV: Extensions * Sectionally Complemented Lattices * Semimodular Lattices * Isoform Lattices * Independence Theorems * Magic Wands Part V: Two Lattices * Sublattices * Ideals * Tensor Extensions * Bibliography * Index