General Lattice Theory
Second edition
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Beschreibung
From the first edition: "This book combines the techniques of an introductory text with those of a monograph to introduce the general reader to lattice theory and to bring the expert up to date on the most recent developments. The first chapter, along with a selection of topics from later chapters, can serve as an introductory course covering first concepts, distributive, modular, semimodular, and geometric lattices, and so on. About 900 exercises and almost 130 diagrams help the beginner to learn the basic results and important techniques. The latter parts of each chapter give deeper developments of the fields mentioned above and there are chapters on equational classes (varieties) and free products. More advanced readers will find the almost 200 research problems, the extensive bibliography, and the further topics and references at the end of each chapter of special use."
In this present edition, the work has been significantly updated and expanded. It contains an extensive new bibliography of 530 items and has been supplemented by eight appendices authored by an exceptional group of experts. The first appendix, written by the author, briefly reviews developments in lattice theory, specifically, the major results of the last 20 years and solutions of the problems proposed in the first edition. The other subjects concern distributive lattices and duality (Brian A. Davey and Hilary A. Priestley), continuous geometries (Friedrich Wehrung), projective lattice geometries (Marcus Greferath and Stefan E. Schmidt), varieties (Peter Jipsen and Henry Rose), free lattices (Ralph Freese), formal concept analysis (Bernhard Ganter and Rudolf Wille), and congruence lattices (Thomas Schmidt in collaboration with the author).
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.
Autor*in
B.A. Davey
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"…Grätzer’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
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