Review of Free Lattice from Math. Reviews

96c:06013 06B25 06-02 06-04 06B20 68Q25
Ralph Freese, Jaroslav Je\v zek, J. B. Nation
Free lattices.
Mathematical Surveys and Monographs, 42.
American Mathematical Society, Providence, RI, 1995. viii+293 pp. $65.00. ISBN 0-8218-0389-1
Reviewer: T. S. Blyth.

This scholarly text covers the fascinating subject of free lattices, the general structure of which is very complex. Beautifully set using AmSLaTeX, 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 authors. In the introduction we are treated to an excellent history of the subject, a survey of the applications of the computer algorithms, and pathway guides for readers of various persuasions. As to the extent of the coverage, this is best judged by the list of contents. Chapter 1: "Whitman's solution to the word problem". Basic concepts, free lattices, canonical forms, continuity, fixed point free polynomials and incompleteness, sublattices of free lattices. Chapter 2: "Bounded homomorphisms and related concepts". Bounded homomorphisms, continuity, doubling and congruences on a finite lattice, a refinement of the D relation, semidistributive lattices, splitting lattices, Day's theorem: free lattices are weakly atomic, applications to congruence varieties. Chapter 3: "Covers in free lattices". Elementary theorems on covers in FL(X), J-closed sets and the standard epimorphism, finite lower bounded lattices, the lattice $L\spcheck(w)$, syntactic algorithms, examples, connected components and the bottom of FL(X). Chapter 4: "Day's theorem revisited". Chapter 5: "Sublattices of free lattices and projective lattices". Projective lattices, the free lattice generated by an ordered set, finite sublattices of free lattices, related topics, finite subdirectly irreducible sublattices of free lattices, summary. Chapter 6: "Totally atomic elements". Characterisation, canonical form of kappa of a totally atomic element, the role of totally atomic elements. Chapter 7: "Finite intervals and connected components". Chains of covers, finite intervals, three-element intervals, connected components. Chapter 8: "Singular and semi-singular elements". Chapter 9: "Tschantz's theorem and maximal chains". Chapter 10: "Infinite intervals". Tschantz triples, join irreducible elements that are not canonical joinands, splittings of a free lattice. Chapter 11: "Computational aspects of lattice theory". Preliminaries, ordered sets, finite lattices, representations and contexts, congruence lattices of finite lattices, bounded homomorphisms and splitting lattices, antichains and chain partitions of ordered sets, algorithms for free lattices, finitely presented lattices, diagrams. Chapter 12: "Term rewrite systems and varieties of lattices". Term rewrite systems, no AC TRS (associative and commutative term rewrite system) for lattice theory, an extension, the variety generated by $L\spcheck (w)$, a lattice variety with AC TRS, more varieties with AC TRS.

The book ends with a collection of 14 challenging open problems and a useful bibliography containing 137 items. Copyright American Mathematical Society 1996