General Lattice Theory: Second edition
Springer Science & Business Media, 21.11.2002 - 663 Seiten
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.
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Chapter I First Concepts
Chapter II Distributive Lattices
Chapter III Congruences and Ideals
Chapter IV Modular and Semimodular Lattices
Chapter V Varieties of Lattices
Chapter VI Free Products
Appendix B Distributive Lattices and Duality
Appendix C Congruence Lattices
Appendix D Continuous Geometry
Appendix E Projective Lattice Geometries
Appendix F Varieties of Lattices
Appendix G Free Lattices
Appendix H Applied Lattice Theory Formal Concept Analysis
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algebras applied assume atoms Boolean algebras Boolean lattice bounded called characterization closed compact complemented complete concept congruence lattice congruence relation consider construction contains converse Corollary covers define Definition denote described determined direct distributive lattice dual duality elements embedded equivalent example Exercise exists extension fact Figure Find finite lattice free lattice free product function geometry given gives Grätzer hence holds homomorphism ideal identity implies infinite interval isomorphic join join-irreducible Jónsson lattice L lattice theory Lemma length Math maximal modular lattice natural Note Observe obtain obviously operations polynomials poset prime ideal Problem projective Proof proper Property Prove relative representation represented respectively result ring satisfying Show space statement Stone subdirectly irreducible sublattice subset Theorem topological unique variety variety of lattices verify
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