基本説明
New in softcover. Hardcover was published in 1998. Textbook.
Full Description
In 20 years, tremendous progress has been made in Lattice Theory. Nevertheless, the change is in the superstructure not in the foundation. Accordingly, I decided to leave the book unchanged and add appendices to record the change. In the first appendix: Retrospective, I briefly review developments from the point of view of this book, specifically, the major results of the last 20 years and solutions of the problems proposed in this book. It is remarkable how many difficult problems have been solved! I was lucky in getting an exceptional group of people to write the other appendices: Brian A. Davey and Hilary A. Priestley on distributive lattices and duality, Friedrich Wehrung on continuous geometries, Marcus Greferath and Stefan E. Schmidt on projective lattice geometries, Peter Jipsen and Henry Rose on varieties, Ralph Freese on free lattices, Bernhard Ganter and Rudolf Wille on formal concept analysis; Thomas Schmidt collaborated with me on congruence lattices. Many of these same people are responsible for the definitive books on the same subjects. I changed very little in the book proper. The diagrams have been redrawn and the book was typeset in ‾1EX. To bring the notation up-to-date, I substituted ConL for C(L), IdL for I(L), and so on. Almost 200 mathematicians helped me with this project, from correcting typos to writing long essays on the topics that should go into Retrospective. The last section of Retrospective lists the major contributors. My deeply felt thanks to all of them.
Contents
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.
