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Lattice Theory: First Concepts and Distributive Lattices

George Grätzer
Publisher: 
Dover Publications
Publication Date: 
2009
Number of Pages: 
212
Format: 
Paperback
Price: 
9.95
ISBN: 
9780486471730
Category: 
Monograph
[Reviewed by
Leon Harkleroad
, on
09/25/2009
]

Often a reprint edition performs the service of rescuing a book from oblivion. But George Grätzer’s Lattice Theory has not really gone away, making this new incarnation somewhat superfluous.

The preface to the original 1971 Lattice Theory mentioned a “companion volume” on which Grätzer was already working. In 1978 General Lattice Theory appeared. This new book of six chapters incorporated most of the material of its predecessor into the first two chapters.

By this century, the field had progressed considerably and General Lattice Theory was starting to show its age. So in 2003 Birkhäuser published a new edition that contained eight appendices written by Grätzer and other leading specialists. These appendices addressed recent developments in lattice theory, as well as some of its applications. Widely held in high esteem, General Lattice Theory appears on the MAA’s Basic Library List.

In short, the second edition of General Lattice Theory basically includes the original Lattice Theory, but with corrections, a more modern take on things, and much more material. Thus there seems little incentive to obtain a reproduction of the 1971 version.

Some disturbing aspects of the blurb on the back cover of the Dover edition also demand comment. The blurb states that the book “combines the techniques of an introductory text with those of a monograph.” Although this description occurs without attribution, it actually appeared word-for-word (except with “textbook” instead of “text”) in Alexandru Carausu’s review of General Lattice Theory several years ago.

The blurb additionally promises “[e]ight appendixes, contributed by a group of experts.” However, Lattice Theory does not and never did contain such appendices; only General Lattice Theory does. If it is too much to ask that people paid to write blurbs either write their own copy or give appropriate credit when using others’ words, can we at least hope that they describe the right book?


Leon Harkleroad wrote a recent MAA Review of a related book, Introduction to Boolean Algebras, by Givant and Halmos.

 

PREFACE
ACKNOWLEDGMENTS
TABLE OF NOTATION
FIRST CONCEPTS
Two Definitions of Lattices
How to Describe Lattices
Some Algebraic Concepts
Polynomials, Identities, and Inequalities
Free Lattices
Special Elements
Further Topics and References
Problems
DISTRIBUTIVE LATTICES
Characterization Theorems and Representation Theorems
Polynomials and Freeness
Congruence Relations
Boolean Algebras R-generated by Distributive Lattices
Topological Representation
Free Distributive Product
Some Categorical Concepts
Further Topics and References
Problems
DISTRIBUTIVE LATTICES WITH PSEUDOCOMPLEMENTATION
Introduction and Stone Algebras
Identities and Congruences
Representation Theorems
Injective and Free Stone Algebras
Further Topics and References
Problems
BIBLIOGRAPHY
INDEX

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