### 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*