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Free lattices. (English) Zbl 0839.06005

Mathematical Surveys and Monographs. 42. Providence, RI: American Mathematical Society (AMS). viii, 293 p. (1995).
This book presents a thorough treatment of the structure of free lattices. It is a book for the research mathematician or graduate student who is interested in the newest results about free lattices. Many of the results presented in the book are new or have not been published elsewhere. Unsolved problems are encountered throughout the text and a list of open problems has been assembled at the end of the book. The book is well written and has an abundance of well-chosen examples. All results that have been published elsewhere are carefully referenced with an extensive bibliography.
The material is organized into twelve chapters. The first chapter, after introducing basic terminology, presents Whitman’s solution of the word problem for free lattices and related results on canonical form, continuity, fixed point free polynomials, and sublattices of free lattices.
In chapter 2 the authors investigate bounded homomorphisms, splitting lattices, and the fundamental ideas concerning upper and lower bounded lattices and their connections with free lattices and varieties of lattices.
Covers of elements in finitely generated free lattices are the subject of chapter 3. Included is a syntactic algorithm to decide if a join-irreducible element of a free lattice is completely join-irreducible. The results of chapter 3 lead to a constructive proof of A. Day’s theorem that finitely generated free lattices are weakly atomic (chapter 4).
Sublattices of free lattices and projective lattices are studied in chapter 5, and totally atomic elements, which play an important role in determining the fine structure of finitely generated free lattices are the subject of chapter 6. Chapters 7-9 contain results on finite intervals, chains of covers and connected components in finitely generated free lattices and develop the machinery that will lead to a very lengthy proof of the fact that there are no Tschantz triples in such lattices, confirming a conjecture by Tschantz on infinite intervals in free lattices.
In chapter 11 the authors have assembled a library of algorithms that are important in the study of ordered sets and lattices. One goal here is to analyze the efficiency of the algorithms, another is to present the algorithms in a manner so that the reader can implement them. Included are algorithms for finding linear extensions, chain partitions, and maximum sized antichains, for determining if a finite lattice is a splitting lattice and for testing if a lattice is simple.
A chapter on term rewrite systems and varieties of lattices concludes the monograph.
Reviewer: M.Höft (Dearborn)

MSC:

06B25 Free lattices, projective lattices, word problems
06-02 Research exposition (monographs, survey articles) pertaining to ordered structures
06-04 Software, source code, etc. for problems pertaining to ordered structures
68Q25 Analysis of algorithms and problem complexity
06A06 Partial orders, general
06B20 Varieties of lattices
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