This presentation on the basics of Boolean algebra has ranked among the fundamental books on this important subject in mathematics and computing science since its initial publication in 1963. Concise and informal as well as systematic, the text draws upon lectures delivered by Professor Halmos at the University of Chicago to cover many topics in brief individual chapters. The approach is suitable for advanced undergraduates and graduate students in mathematics. Starting with Boolean rings and algebras, the treatment examines fields of sets, regular open sets, elementary relations, infinite operations, subalgebras, homomorphisms, free algebras, ideals and filters, and the homomorphism theorem. Additional topics include measure algebras, Boolean spaces, the representation theorem, duality for ideals and for homomorphisms, Boolean measure spaces, isomorphisms of factors, projective and injective algebras, and many other subjects. Several chapters conclude with stimulating exercises; the solutions are not included.

IN 1959 I lectured on Boolean algebras at the University of Chicago. A mimeographed version of the notes on which the lectures were based circulated for about two years; this volume contains those notes, corrected and revised. Most of the corrections were suggested by Peter Crawley. To judge by his detailed and precise suggestions, he must have read every word, checked every reference, and weighed every argument, and I am lIery grateful to hirn for his help. This is not to say that he is to be held responsible for the imperfec tions that remain, and, in particular, I alone am responsible for all expressions of personal opinion and irreverent view point. P. R. H. Ann Arbor, Michigan ] anuary, 1963 Contents Section Page 1 1 Boolean rings ............................ . 2 Boolean algebras ......................... . 3 9 3 Fields of sets ............................ . 4 Regular open sets . . . . . . . . . . . . . . . . . . . 12 . . . . . . 5 Elementary relations. . . . . . . . . . . . . . . . . . 17 . . . . . 6 Order. . . . . . . . . . . . . . . . . . . . . . . . . . . 21 . . . . . . . . . 7 Infinite operations. . .. . . . . . . . . . . . . . . . . 25 . . . . . 8 Subalgebras . . . . . . . . . . . . . . . . . . . . .. . . . 31 . . . . . . 9 Homomorphisms . . . . . . . . . . . . . . . . . . . . 35 . . . . . . . 10 Free algebras . . . . . . . . . . . . . . . . . . . . . . 40 . . . . . . . 11 Ideals and filters. . . . . . . . . . . . . . . . . . . . 47 . . . . . . 12 The homomorphism theorem. . . . . . . . . . . . .. . . 52 . . 13 Boolean a-algebras . . . . . . . . . . . . . . . . . . 55 . . . . . . 14 The countable chain condition . . . . . . . . . . . . 61 . . . 15 Measure algebras . . . . . . . . . . . . . . . . . . . 64 . . . . . . . 16 Atoms.. . . . .. . . . . .. .. . . . ... . . . . .. . . ... . . .. 69 17 Boolean spaces . . . . . . . . . . . . . . . . . . . . 72 . . . . . . . 18 The representation theorem. . . . . . . . . . . . . . 77 . . . 19 Duali ty for ideals . . . . . . . . . . . . . . . . . .. . . 81 . . . . . 20 Duality for homomorphisms . . . . . . . . . . . . . . 84 . . . . 21 Completion . . . . . . . . . . . . . . . . . . . . . . . 90 . . . . . . . . 22 Boolean a-spaces . . . . . . . . . . . . . . . . . .. . . 97 . . . . . 23 The representation of a-algebras . . . . . . . . .. . . 100 . 24 Boolean measure spaces . . . . . . . . . . . . . .. . . 104 . . . 25 Incomplete algebras . . . . . . . . . . . . . . . .. . . 109 . . . . . 26 Products of algebras . . . . . . . . . . . . . . . .. . . 115 . . . . 27 Sums of algebras . . . . . . . . . . . . . . . . . .. . . 119 . . . . . 28 Isomorphisms of factors . . . . . . . . . . . . . .. . . 122 . . .

This book is an informal though systematic series of lectures on Boolean algebras. It contains background chapters on topology and continuous functions and includes hundreds of exercises as well as a solutions manual.

There are two aspects to the theory of Boolean algebras; the algebraic and the set-theoretical. A Boolean algebra can be considered as a special kind of algebraic ring, or as a generalization of the set-theoretical notion of a field of sets. Fundamental theorems in both of these directions are due to M. H. STONE, whose papers have opened a new era in the develop ment of this theory. This work treats the set-theoretical aspect, with little mention being made of the algebraic one. The book is composed of two chapters and an appendix. Chapter I is devoted to the study of Boolean algebras from the point of view of finite Boolean operations only; a greater part of its contents can be found in the books of BIRKHOFF [2J and HERMES [IJ. Chapter II seems to be the first systematic study of Boolean algebras with infinite Boolean operations. To understand Chapters I and II it suffices only to know fundamental notions from general set theory and set-theoretical topology. No know ledge of lattice theory or of abstract algebra is presumed. Less familiar topological theorems are recalled, and only a few examples use more advanced topological means; but these may be omitted. All theorems in both chapters are given with full proofs.

Concise and informal as well as systematic, this presentation on the basics of Boolean algebra has ranked among the fundamental books on the subject since its initial publication in 1963.