General Lattice Theory

"...Gratzer's 'General Lattice Theory' has become the lattice theorist's bible. Now...we have the second edition, in which the old testament is augmented by a new testament... The new testament gospel is provided by leading and acknowledged experts in their fields... Each [of these eight contributions] is itself a gold mine. This is an excellent and engaging second edition that will long remain a standard reference." -MATHEMATICAL REVIEWS "Despite the large number of coauthors the style is uniform and the book is well written. As the first edition of this book had a deep influence on the development of lattice theory, I expect that the new edition will continue to hold its leading position among the books on lattice theory." -ZENTRALBLATT MATH "This second edition of the Gratzer's book on lattice theory is an expanded and updated form of its first edition. Following the line of first edition, it combines the techniques of an introductory textbook with those of a monograph to introduce the reader to lattice theory and to bring the expert up to date on the most recent developments... Author adds eight appendices to record the changes in the superstructure of lattice theory that occurred in the time between the two editions of this book. In the first appendix, the authro reviews the major results of the last 20 years and solutions of the problems proposed in this book... Almost 900 exercises form an important part of this book. The bibliography contains over 750 entries. A very detailed index and the Table of Notation should help the reader in finding where a concept or notation was first introduced." ---ANALELE STIINTIFICE ALE UNIVERSITATII

• I First Concepts.- 1 Two Definitions of Lattices.- 2 How to Describe Lattices.- 3 Some Algebraic Concepts.- 4 Polynomials, Identities, and Inequalities.- 5 Free Lattices.- 6 Special Elements.- Further Topics and References.- Problems.- II Distributive Lattices.- 1 Characterization and Representation Theorems.- 2 Polynomials and Freeness.- 3 Congruence Relations.- 4 Boolean Algebras.- 5 Topological Representation.- 6 Pseudocomplementation.- Further Topics and References.- Problems.- III Congruences and Ideals.- 1 Weak Projectivity and Congruences.- 2 Distributive, Standard, and Neutral Elements.- 3 Distributive, Standard, and Neutral Ideals.- 4 Structure Theorems.- Further Topics and References.- Problems.- IV Modular and Semimodular Lattices.- 1 Modular Lattices.- 2 Semimodular Lattices.- 3 Geometric Lattices.- 4 Partition Lattices.- 5 Complemented Modular Lattices.- Further Topics and References.- Problems.- V Varieties of Lattices.- 1 Characterizations of Varieties.- 2 The Lattice of Varieties of Lattices.- 3 Finding Equational Bases.- 4 The Amalgamation Property.- Further Topics and References.- Problems.- VI Free Products.- 1 Free Products of Lattices.- 2 The Structure of Free Lattices.- 3 Reduced Free Products.- 4 Hopfian Lattices.- Further Topics and References.- Problems.- Concluding Remarks.- Table of Notation.- A Retrospective.- 1 Major Advances.- 2 Notes on Chapter I.- 3 Notes on Chapter II.- 4 Notes on Chapter III.- 5 Notes on Chapter IV.- 6 Notes on Chapter V.- 7 Notes on Chapter VI.- 8 Lattices and Universal Algebras.- B Distributive Lattices and Duality by B. Davey, II. Priestley.- 1 Introduction.- 2 Basic Duality.- 3 Distributive Lattices with Additional Operations.- 4 Distributive Lattices with V-preserving Operators, and Beyond.- 5 The Natural Perspective.- 6 Congruence Properties.- 7 Freeness, Coproducts, and Injectivity.- C Congruence Lattices by G. Gratzer, E. T. Schmidt.- 1 The Finite Case.- 2 The General Case.- 3 Complete Congruences.- D Continuous Geometry by F. Wehrung.- 1 The von Neumann Coordinatization Theorem.- 2 Continuous Geometries and Related Topics.- E Projective Lattice Geometries by M. Greferath, S. Schmidt.- 1 Background.- 2 A Unified Approach to Lattice Geometry.- 3 Residuated Maps.- F Varieties of Lattices by P. Jipsen, H. Rose.- 1 The Lattice A.- 2 Generating Sets of Varieties.- 3 Equational Bases.- 4 Amalgamation and Absolute Retracts.- 5 Congruence Varieties.- G Free Lattices by R. Frecse.- 1 Whitman’s Solutions; Basic Results.- 2 Classical Results.- 3 Covers in Free Lattices.- 4 Semisingular Elements and Tschantz’s Theorem.- 5 Applications and Related Areas.- H Formal Concept Analysis by B. Cantor and R. Wille.- 1 Formal Contexts and Concept Lattices.- 2 Applications.- 3 Sublattices and Quotient Lattices.- 4 Subdirect Products and Tensor Products.- 5 Lattice Properties.- New Bibliography.