eBook ISBN:  9780821834008 
Product Code:  CONM/76.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
eBook ISBN:  9780821834008 
Product Code:  CONM/76.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 

Book DetailsContemporary MathematicsVolume: 76; 1988; 203 ppMSC: Primary 08; Secondary 03
The utility of congruence lattices in revealing the structure of general algebras has been recognized since Garrett Birkhoff's pioneering work in the 1930s and 1940s. However, the results presented in this book are of very recent origin: most of them were developed in 1983. The main discovery presented here is that the lattice of congruences of a finite algebra is deeply connected to the structure of that algebra. The theory reveals a sharp division of locally finite varieties of algebras into six interesting new families, each of which is characterized by the behavior of congruences in the algebras. The authors use the theory to derive many new results that will be of interest not only to universal algebraists, but to other algebraists as well.
The authors begin with a straightforward and complete development of basic tame congruence theory, a topic that offers great promise for a wide variety of investigations. They then move beyond the consideration of individual algebras to a study of locally finite varieties. A list of open problems closes the work.

Table of Contents

Chapters

Introduction

Chapter 0: Basic concepts and notation

Chapter 1: Tight lattices

Chapter 2: Tame quotients

Chapter 3: Abelian and solvable algebras

Chapter 4: The structure of minimal algebras

Chapter 5: The types of tame quotients

Chapter 6: Labeled congruence lattices

Chapter 7: Solvability and semidistributivity

Chapter 8: Congruence modular varieties

Chapter 9: Mal$\prime $cev classification and omitting types

Chapter 10: Residually small varieties

Chapter 11: Decidable varieties

Chapter 12: Free spectra

Chapter 13: Tame algebras and Eminimal algebras

Chapter 14: Simple algebras in varieties

Problems

An appendix added in July, 1996

Bibliography

Added in July, 1996

Index to Terms

Index of Notation


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The utility of congruence lattices in revealing the structure of general algebras has been recognized since Garrett Birkhoff's pioneering work in the 1930s and 1940s. However, the results presented in this book are of very recent origin: most of them were developed in 1983. The main discovery presented here is that the lattice of congruences of a finite algebra is deeply connected to the structure of that algebra. The theory reveals a sharp division of locally finite varieties of algebras into six interesting new families, each of which is characterized by the behavior of congruences in the algebras. The authors use the theory to derive many new results that will be of interest not only to universal algebraists, but to other algebraists as well.
The authors begin with a straightforward and complete development of basic tame congruence theory, a topic that offers great promise for a wide variety of investigations. They then move beyond the consideration of individual algebras to a study of locally finite varieties. A list of open problems closes the work.

Chapters

Introduction

Chapter 0: Basic concepts and notation

Chapter 1: Tight lattices

Chapter 2: Tame quotients

Chapter 3: Abelian and solvable algebras

Chapter 4: The structure of minimal algebras

Chapter 5: The types of tame quotients

Chapter 6: Labeled congruence lattices

Chapter 7: Solvability and semidistributivity

Chapter 8: Congruence modular varieties

Chapter 9: Mal$\prime $cev classification and omitting types

Chapter 10: Residually small varieties

Chapter 11: Decidable varieties

Chapter 12: Free spectra

Chapter 13: Tame algebras and Eminimal algebras

Chapter 14: Simple algebras in varieties

Problems

An appendix added in July, 1996

Bibliography

Added in July, 1996

Index to Terms

Index of Notation