简介
In this present edition, the work has been significantly updated and expanded. It contains an extensive new bibliography of 530 items and has been supplemented by eight appendices authored by an exceptional group of experts. The first appendix, written by the author, briefly reviews developments in lattice theory, specifically, the major results of the last 20 years and solutions of the problems proposed in the first edition. The other subjects concern distributive lattices and duality (Brian A. Davey and Hilary A. Priestley), continuous geometries (Friedrich Wehrung), projective lattice geometries (Marcus Greferath and Stefan E. Schmidt), varieties (Peter Jipsen and Henry Rose), free lattices (Ralph Freese), formal concept analysis (Bernhard Ganter and Rudolf Wille), and congruence lattices (Thomas Schmidt in collaboration with the author).
目录
Front Cover 1
General Lattice Theory 4
Copyright Page 5
Contents 8
Preface and Acknowledgements 10
Introduction 12
CHAPTER I. FIRST CONCEPTS 16
1. Two Definitions of Lattices 16
2. How to Describe Lattices 24
3. Some Algebraic Concepts 30
4. Polynomials, Identities, and Inequalities 41
5. Free Lattices 47
6. Special Elements 62
Further Topics and References 67
Problems 71
CHAPTER II. DISTRIBUTIVE LATTICES 74
1. Characterization Theorems and Representation Theorems 74
2. Polynomials and Freeness 153
3. Congruence Relations 88
4. Boolean Algebras R-generated by Distributive Lattices 101
5. Topological Representation 114
6. Distributive Lattices with Pseudocomplementation 126
Further Topics and References 135
Problems 141
CHAPTER III. CONGRUENCES AND IDEALS 144
1. Weak Projectivity and Congruences 144
2. Distributive, Standard, and Neutral Elements 153
3. Distributive, Standard, and Neutral Ideals 161
4. Structure Theorems 166
Further Topics and References 173
Problems 174
CHAPTER IV. MODULAR AND SEMIMODULAR LATTICES 176
1. Modular Lattices 176
2. Semimodular Lattices 187
3. Geometric Lattices 193
4. Partition Lattices 207
5. Complemented Modular Lattices 216
Further Topics and References 233
Problems 239
CHAPTER V. EQUATIONAL CLASSES OF LATTICES 242
1. Characterizations of Equational Classes 242
2. The Lattice of Equational Classes of Lattices 251
3. Finding Equational Bases 258
4. The Amalgamation Property 267
Further Topics and References 275
Problems 277
CHAPTER VI. FREE PRODUCTS 280
1. Free Products of Lattices 280
2. The Structure of Free Lattices 297
3. Reduced Free Products 303
4. Hopfien Lattices 313
Further Topics and References 318
Problems 321
CONCLUDING REMARKS 326
BIBLIOGRAPHY 331
TABLE OF NOTATION 377
INDEX 380
General Lattice Theory 4
Copyright Page 5
Contents 8
Preface and Acknowledgements 10
Introduction 12
CHAPTER I. FIRST CONCEPTS 16
1. Two Definitions of Lattices 16
2. How to Describe Lattices 24
3. Some Algebraic Concepts 30
4. Polynomials, Identities, and Inequalities 41
5. Free Lattices 47
6. Special Elements 62
Further Topics and References 67
Problems 71
CHAPTER II. DISTRIBUTIVE LATTICES 74
1. Characterization Theorems and Representation Theorems 74
2. Polynomials and Freeness 153
3. Congruence Relations 88
4. Boolean Algebras R-generated by Distributive Lattices 101
5. Topological Representation 114
6. Distributive Lattices with Pseudocomplementation 126
Further Topics and References 135
Problems 141
CHAPTER III. CONGRUENCES AND IDEALS 144
1. Weak Projectivity and Congruences 144
2. Distributive, Standard, and Neutral Elements 153
3. Distributive, Standard, and Neutral Ideals 161
4. Structure Theorems 166
Further Topics and References 173
Problems 174
CHAPTER IV. MODULAR AND SEMIMODULAR LATTICES 176
1. Modular Lattices 176
2. Semimodular Lattices 187
3. Geometric Lattices 193
4. Partition Lattices 207
5. Complemented Modular Lattices 216
Further Topics and References 233
Problems 239
CHAPTER V. EQUATIONAL CLASSES OF LATTICES 242
1. Characterizations of Equational Classes 242
2. The Lattice of Equational Classes of Lattices 251
3. Finding Equational Bases 258
4. The Amalgamation Property 267
Further Topics and References 275
Problems 277
CHAPTER VI. FREE PRODUCTS 280
1. Free Products of Lattices 280
2. The Structure of Free Lattices 297
3. Reduced Free Products 303
4. Hopfien Lattices 313
Further Topics and References 318
Problems 321
CONCLUDING REMARKS 326
BIBLIOGRAPHY 331
TABLE OF NOTATION 377
INDEX 380
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