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Publisher:

Dover Publications

Publication Date:

2009

Number of Pages:

212

Format:

Paperback

Price:

9.95

ISBN:

9780486471730

Category:

Monograph

[Reviewed by , on ]

Leon Harkleroad

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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