Lattice theory / Garrett Birkhoff

Bibliogr. Index

This expanded and thoroughly reorganized edition contains accounts of many important discoveries of the last twenty years, integrated with a streamlined and modernized treatment of the material of the previous editions. It amply attests to the growth and vitality of the subject during those years. With the size of the volume, the length of the list of unsolved problems continues to grow—166 in this edition. Both suggest that lattice theory will continue to be the subject of increasing research activity.
In Chapters I-V the various types of lattices are introduced and studied. Chapter III deals with structure and representation theory, including new work of Grätzer and Schmidt, and Chapter IV with geometric lattices, including recent discoveries of Rota and others. Applications to algebra are considered in Chapters VI and VII; such topics as free algebras, the word problem, group theory, permutable congruence relations, and the structure lattice are treated.
In Chapters VIII-XII applications to set theory and analysis (including topology and measure theory) are discussed. Chapter VIII opens with a treatment of transfinite induction and related matters, and proceeds to a proof of the subdirect decomposition theorem. Algebraic lattices are introduced and studied (among them the subalgebra and structure lattices of an algebra). Up to this point the book is mainly concerned with lattices of finite length and their generalization, the algebraic lattices. Chapters IX-XI deal with continuous lattices. In Chapter IX lattices of open (or closed) sets are considered, leading to a lattice-theoretic treatment of compactness and to Stone's representation theorem. Chapter X develops the theory of metric and topological lattices, including continuous geometries, and Chapter XI Borel and measure algebras, von Neumann lattices, and dimension theory.
The remainder of the book is devoted to partially ordered systems having an additional binary operation. The theory of lattice-ordered groups is systematically developed, as is that of vector lattices, both chapters containing much material new in this edition. There is a chapter on lattice-ordered monoids, with applications to the ideal theory of Noetherian rings and to relation algebras. An interesting chapter on positive linear operators includes various generalizations of Perron's theorem on positive matrices, as well as treatments of transition operators and ergodic theory. The final chapter introduces lattice-ordered rings, and contains a section on averaging operators. (MathSciNet)