Lattice Theory: Foundation

Unfortunately, the word “lattice” has more than one mathematical meaning. For the record, Grätzer’s book does not deal with structures such as the Gaussian integers or the Leech lattice. Rather, it concerns itself with sets in which every pair of elements possesses a sup and an inf. Lattices in this sense arise in a wide variety of mathematical contexts. For example, the set of normal subgroups of a group, the set of ideals of a commutative ring, and the set of subspaces of a vector space all form lattices — in fact, so-called modular lattices, which enjoy a distributivity-like property connecting sup and inf.

Dedekind’s seminal study of algebraic properties of lattices grew out of his number-theoretic investigations. Likewise, Von Neumann’s work on operator rings in Hilbert spaces led him to a theory of a particular kind of lattice, known as a continuous geometry. And of course, Boolean algebras represent yet another important class of lattices. But despite the ubiquity, history, and rich theory of lattices in this sense, they often play second fiddle to the other kind of lattice. For example, even in the excellent and comprehensive Princeton Companion to Mathematics, not one of the index’s twelve listings under “lattices” refers to the sup-inf type.

So where can one turn to learn about lattices in the sup-inf sense? Lattice Theory: Foundation represents the latest stage in the evolution of a classic in the field, which Grätzer has periodically revamped and rewritten over the decades in order to keep pace with the multitude of advances that have occurred. Quoting from a recent MAA Review of a reprint of the primordial ancestor of this book, “The preface to the original 1971 Lattice Theory mentioned a `companion volume’ on which Grätzer was already working. In 1978 General Lattice Theory [GLT] appeared. This new book of six chapters incorporated most of the material of its predecessor into the first two chapters. By this century, the field had progressed considerably and GLT was starting to show its age. So in 2003 Birkhäuser published a new edition that contained eight appendices written by Grätzer and other leading specialists. These appendices addressed recent developments in lattice theory, as well as some of its applications. Widely held in high esteem, GLT appears on the MAA’s Basic Library List.”

The area has continued to progress, so that Grätzer is now splitting the book into two volumes. In his words, “To lay the foundation, to survey the contemporary field, to pose research problems, would require more than one volume or more than one person. So I decided to cut back and concentrate in this volume on the foundation. … My plan was to revise, reorganize, and up-to-date the old chapters, add the foundation for congruence lattices of finite lattices (lattice constructions), and modernize notation.”

Thus the new book roughly corresponds to the core chapters of GLT, and the projected companion volume Lattice Theory: Topics and Applications to the appendices of GLT. As suggested by Grätzer’s comments, though, Lattice Theory: Foundation contains many changes from GLT. The material on lattice constructions that he mentioned constitutes a new 50-page chapter, including techniques of comparatively recent vintage.

Also added to the book are 44 passages, flagged with a diamond label, typically a page or so in length. These sections, 16 of them contributed by guest authors, “provide brief glimpses into research fields beyond the horizon of this book,” covering topics such as Dilworth’s covering theorem and the word problem for modular lattices. As in previous incarnations, the text includes a large number of exercises at a variety of levels of difficulty. The lists of open research problems, however, have been dropped from this volume.

The sup-inf type of lattice may sometimes keep a lower profile than its periodic namesake, but Grätzer has certainly given it its due, providing a thorough account of the fundamentals. Here’s hoping that the companion volume appears soon.

Leon Harkleroad wishes to thank the author of the MAA Review cited above for kindly granting permission to quote it here.