**Mathematical Surveys and Monographs**

Volume: 42;
1995;
293 pp;
Hardcover

MSC: Primary 06;
Secondary 68

Print ISBN: 978-0-8218-0389-9

Product Code: SURV/42

List Price: $83.00

Individual Member Price: $66.40

**Electronic ISBN: 978-1-4704-1273-9
Product Code: SURV/42.E**

List Price: $83.00

Individual Member Price: $66.40

# Free Lattices

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*Ralph Freese; Jaroslav Ježek; J. B. Nation*

This book explores the surprisingly rich and complex structure of free lattices. The first part of the book presents a complete exposition of the basic theory of free lattices, projective lattices, and lattices which are bounded homomorphic images of a free lattice, as well as applications of these results to other areas. This portion of the book is suitable for use in a graduate course in lattice theory or general algebra. The second part of the book contains new results about free lattices and new proofs of known results, providing the reader with a coherent picture of the fine structure of free lattices. The book closes with an analysis of algorithms for free lattices and finite lattices that is accessible to researchers in other areas and depends only on the first chapter and a small part of the second. Several open problems appear throughout the book and, for easy reference, are assembled in a section at the end. Synthesizing seventy years of research, this is the only comprehensive treatment available of the theory of free lattices.

#### Readership

Mathematicians interested in algebra and lattice theory in particular. Mathematicians with an interest in algorithms in lattice theory and their complexities.

#### Reviews & Endorsements

Researchers in lattice theory will find much of interest. The comprehensive nature of this book guarantees that those interested in further research in lattices will find this book indispensable.

An extensive body of results on free lattices exists, and the twelve chapters of this book provide a comprehensive yet engaging survey of these results … Anyone working in lattice theory or in the study of free algebraic systems should certainly have access to a copy.

-- Bulletin of the AMS

This scholarly text covers the fascinating subject of free lattices, the general structure of which is very complex. Beautifully set using TeX, and reasonably priced, it provides the lattice theorist with a wealth of information on the subject and is sure to become a classic reference. The reader can be assured of the quality of the exposition simply from the names of the author … The book ends with a collection of 14 challenging open problems and a useful bibliography.

-- Mathematical Reviews

Well written and has an abundance of well-chosen examples.

-- Zentralblatt MATH

An extensive body of results on free lattices exists, and the twelve chapters of this book provide a comprehensive yet engaging survey of these results … Anyone working in lattice theory or in the study of free algebraic systems should certainly have access to a copy.

-- Bulletin of the London Mathematical Society

A beautiful book which collects all what is known on free lattices!

-- Monatshefte für Mathematik

#### Table of Contents

# Table of Contents

## Free Lattices

- Contents vii8 free
- Introduction 110 free
- Chapter I. Whitman's Solution to the Word Problem 716 free
- Chapter II. Bounded Homomorphisms and Related Concepts 2736
- 1. Bounded Homomorphisms 2736
- 2. Continuity 3746
- 3. Doubling and Congruences on a Finite Lattice 3847
- 4. A Refinement of the D Relation 4554
- 5. Semidistributive Lattices 4958
- 6. Splitting Lattices 5564
- 7. Day's Theorem: Free Lattices are Weakly Atomic 6271
- 8. Applications to Congruence Varieties 6574

- Chapter III. Covers in Free Lattices 6776
- Chapter IV. Day's Theorem Revisited 91100
- Chapter V. Sublattices of Free Lattices and Projective Lattices 95104
- Chapter VI. Totally Atomic Elements 135144
- Chapter VII. Finite Intervals and Connected Components 151160
- Chapter VIII. Singular and Semisingular Elements 171180
- Chapter IX. Tschantz's Theorem and Maximal Chains 179188
- Chapter X. Infinite Intervals 189198
- Chapter XI. Computational Aspects of Lattice Theory 205214
- 1. Preliminaries 205214
- 2. Ordered Sets 208217
- 3. Finite Lattices 215224
- 4. Representations and Contexts 220229
- 5. Congruence Lattices of Finite Lattices 221230
- 6. Bounded Homomorphisms and Splitting Lattices 229238
- 7. Antichains and Chain Partitions of Ordered Sets 235244
- 8. Algorithms for Free Lattices 241250
- 9. Finitely Presented Lattices 249258
- 10. Diagrams 251260

- Chapter XII. Term Rewrite Systems and Varieties of Lattices 255264
- Open Problems 279288
- Bibliography 281290
- List of Symbols 287296
- Index 289298