Lattice Theory: Foundation

George Grätzer, 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.

From the reviews:

"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 Grätzer'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." Garrett Birkhoff (Bulletin of the American Mathematical Society

"Grätzer's book General Lattice Theory has become the lattice theorist's bible." (Mathematical Reviews)

"This book has a long history maturing in contents and details over a period of more than 40 years. ... For many years to come it will be the standard textbook and main reference for everybody interested in Lattice Theory. ... It contains more than 1,000 exercises, a bibliography of 742 items, and a useful glossary of notation. All this, together with the very clear style of presentation, makes the book an extremely valuable contribution to the theory of lattices." (M. Mitsch, Monatshefte für Mathematik, Vol. 164 (4), 2011)

"This book is a new version of the author's famous book on lattice theory, which appeared in several editions ... . The bibliography was updated and now contains 742 items. ... the integration of the former appendices into the main body of the book was a very good idea, as it will help the reader to get a clearer picture of the current state of the art in lattice theory." (G. Eigenthaler, Zentralblatt MATH, Vol. 1233, 2012)