Introduction to Boolean algebras

Paul Halmos, Steven Givant (auth.)0387402934, 9780387402932, 9780387684369, 0387684360

In a bold and refreshingly informal style, this exciting text steers a middle course between elementary texts emphasizing connections with philosophy, logic, and electronic circuit design, and profound treatises aimed at advanced graduate students and professional mathematicians. It is written for readers who have studied at least two years of college-level mathematics. With carefully crafted prose, lucid explanations, and illuminating insights, it guides students to some of the deeper results of Boolean algebra — and in particular to the important interconnections with topology — without assuming a background in algebra, topology, and set theory. The parts of those subjects that are needed to understand the material are developed within the text itself.

Highlights of the book include the normal form theorem; the homomorphism extension theorem; the isomorphism theorem for countable atomless Boolean algebras; the maximal ideal theorem; the celebrated Stone representation theorem; the existence and uniqueness theorems for canonical extensions and completions; Tarski’s isomorphism of factors theorem for countably complete Boolean algebras, and Hanf’s related counterexamples; and an extensive treatment of the algebraic-topological duality, including the duality between ideals and open sets, homomorphisms and continuous functions, subalgebras and quotient spaces, and direct products and Stone-Cech compactifications.

A special feature of the book is the large number of exercises of varying levels of difficulty, from routine problems that help readers understand the basic definitions and theorems, to intermediate problems that extend or enrich material developed in the text, to harder problems that explore important ideas either not treated in the text, or that go substantially beyond its treatment. Hints for the solutions to the harder problems are given in an appendix. A detailed solutions manual for all exercises is available for instructors who adopt the text for a course.

Table of contents :
Front Matter….Pages 1-14
Boolean Rings….Pages 1-7
Boolean Algebras….Pages 8-13
Boolean Algebras Versus Rings….Pages 14-19
The Principle of Duality….Pages 20-23
Fields of Sets….Pages 24-30
Elementary Relations….Pages 31-37
Order….Pages 38-44
Infinite Operations….Pages 45-52
Topology….Pages 53-65
Regular Open Sets….Pages 66-73
Subalgebras….Pages 74-88
Homomorphisms….Pages 89-104
Extensions of Homomorphisms….Pages 105-116
Atoms….Pages 117-126
Finite Boolean Algebras….Pages 127-133
Atomless Boolean Algebras….Pages 134-141
Congruences and Quotients….Pages 142-148
Ideals and Filters….Pages 149-163
Lattices of Ideals….Pages 164-170
Maximal Ideals….Pages 171-177
Homomorphism and Isomorphism Theorems….Pages 178-187
The Representation Theorem….Pages 188-192
Canonical Extensions….Pages 193-199
Complete Homomorphisms and Complete Ideals….Pages 200-213
Completions….Pages 214-220
Products of Algebras….Pages 221-242
Isomorphisms of Factors….Pages 243-255
Free Algebras….Pages 256-267
Boolean s-algebras….Pages 268-281
The Countable Chain Condition….Pages 282-287
Measure Algebras….Pages 288-299
Boolean Spaces….Pages 300-311
Continuous Functions….Pages 312-325
Boolean Algebras and Boolean Spaces….Pages 326-337
Duality for Ideals….Pages 338-346
Duality for Homomorphisms….Pages 347-358
Duality for Subalgebras….Pages 359-367
Duality for Completeness….Pages 368-372
Boolean s-spaces….Pages 373-377
The Representation of s-algebras….Pages 378-383
Boolean Measure Spaces….Pages 384-389
Incomplete Algebras….Pages 390-395
Duality for Products….Pages 396-421
Sums of Algebras….Pages 422-438
Isomorphisms of Countable Factors….Pages 439-446
Back Matter….Pages 1-128