This book started with Lattice Theory, First Concepts, in 1971. Then came General Lattice Theory, First Edition, in 1978, and the Second Edition twenty years later. Since the publication of the first edition in 1978, General Lattice Theory has become the authoritative introduction to lattice theory for graduate students and the standard reference for researchers. The First Edition set out to introduce and survey lattice theory. Some 12,000 papers have been published in the field since then; so Lattice Theory: Foundation focuses on introducing the field, laying the foundation for special topics and applications. Lattice Theory: Foundation, based on the previous three books, covers the fundamental concepts and results. The main topics are distributivity, congruences, constructions, modularity and semimodularity, varieties, and free products. The chapter on constructions is new, all the other chapters are revised and expanded versions from the earlier volumes. Almost 40 "diamond sections'', many written by leading specialists in these fields, provide a brief glimpse into special topics beyond the basics. "Lattice theory has come a long way... For those who appreciate lattice theory, or who are curious about its techniques and intriguing internal problems, Professor Gratzer's lucid new book provides a most valuable guide to many recent developments. Even a cursory reading should provide those few who may still believe that lattice theory is superficial or naive, with convincing evidence of its technical depth and sophistication." Bulletin of the American Mathematical Society "Gratzer's book General Lattice Theory has become the lattice theorist's bible." Mathematical Reviews
George Gratzer, Member of the Canadian Academy of Sciences and Foreign Member of the Hungarian Academy of Sciences, is the author of 26 books in five languages and about 260 articles, most of them on his research in lattice theory.
Preface.- Introduction.- Glossary of Notation.- I First Concepts.- 1 Two Definitions of Lattices.- 2 How to Describe Lattices.- 3 Some Basic Concepts.- 4 Terms, Identities, and Inequalities.- 5 Free Lattices.- 6 Special Elements.- II Distributive Lattices.- 1 Characterization and Representation Theorems.- 2 Terms and Freeness.- 3 Congruence Relations.- 4 Boolean Algebras.- 5 Topological Representation.- 6 Pseudocomplementation.- III Congruences.- 1 Congruence Spreading.- 2 Distributive, Standard, and Neutral Elements.- 3 Distributive, Standard, and Neutral Ideals.- 4 Structure Theorems.- IV Lattice Constructions.- 1 Adding an Element.- 2 Gluing.- 3 Chopped Lattices.- 4 Constructing Lattices with Given Congruence Lattices.- 5 Boolean Triples.- V Modular and Semimodular Lattices.- 1 Modular Lattices.- 2 Semimodular Lattices.- 3 Geometric Lattices.- 4 Partition Lattices.- 5 Complemented Modular Lattices.- VI Varieties of Lattices.- 1 Characterizations of Varieties 397.- 2 The Lattice of Varieties of Lattices.- 3 Finding Equational Bases.- 4 The Amalgamation Property.- VII Free Products.- 1 Free Products of Lattices.- 2 The Structure of Free Lattices.- 3 Reduced Free Products.- 4 Hopfian Lattices.- Afterword.- Bibliography.