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Free Lattices
Ralph Freese, Jaroslav Ježek, and J. B. Nation
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Mathematical Surveys and Monographs
1995; 293 pp; hardcover
Volume: 42
ISBN-10: 0-8218-0389-1
ISBN-13: 978-0-8218-0389-9
List Price: US$79
Member Price: US$63.20
Order Code: SURV/42
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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

"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

  • Whitman's solution to the word problem
  • Bounded homomorphisms and related concepts
  • Covers in free lattices
  • Day's theorem revisited
  • Sublattices of free lattices and projective lattices
  • Totally atomic elements
  • Finite intervals and connected components
  • Singular and semisingular elements
  • Tschantz's theorem and maximal chains
  • Infinite intervals
  • Computational aspects of lattice theory
  • Term rewrite systems and varieties of lattices
  • Open problems
  • Bibliography
  • List of symbols
  • Index
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