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The theory of Boolean algebras was created in 1847 by the English mathematician George Boole. He conceived it as a calculus (or arithmetic) suitable for a mathematical analysis of logic. The form of his calculus was rather different from the modern version, which came into being during the period 1864–1895 through the contributions of William Stanley Jevons, Augustus De Morgan, Charles Sanders Peirce, and Ernst Schroder. A foundation of the calculus as an abstract algebraic discipline, axiomatized by a set of equations, and admitting many different interpretations, was carried out by Edward Huntington in 1904.
Only with the work of Marshall Stone and Alfred Tarski in the 1930s, however, did Boolean algebra free itself completely from the bonds of logic and become a modern mathematical discipline, with deep theorems and important connections to several other branches of mathematics, including algebra, analysis, logic, measure theory, probability and statistics, set theory, and topology. For instance, in logic, beyond its close connection to propositional logic, Boolean algebra has found applications in such diverse areas as the proof of the completeness theorem for first-order logic, the proof of the _Lo´s conjecture for countable first-order theories categorical in power, and proofs of the independence of the axiom of choice and the continuum hypothesis in set theory. In analysis, Stone’s discoveries of the Stone–Cech compactification and the Stone–Weierstrass approximation theorem were intimately connected to his study of Boolean algebras. Countably complete Boolean algebras (also called σ-algebras) and countably complete fields of sets (also called σ-fields) play a key role in the foundations of measure theory. Outside the realm of mathematics, Boolean algebra has found applications in such diverse areas as anthropology, biology, chemistry, ecology, economics, sociology, and especially computer science and philosophy. For example, in computer science, Boolean algebra is used in electronic circuit design (gating networks), programming languages, databases, and complexity theory. Most books on Boolean algebra fall into one of two categories. There are elementary texts that emphasize the arithmetic aspects of the subject (in particular, the laws that can be expressed and proved in the theory), and that often explore applications to propositional logic, philosophy, and electronic circuit design. There are also advanced treatises that present the deeper mathematical aspects of the theory at a level appropriate for graduate students and professional mathematicians (in terms of the mathematical background and level of sophistication required for understanding the presentation).
This book, a substantially revised version of the second author’s Lectures on Boolean Algebras, tries to steer a middle course. It is aimed at undergraduates who have studied, say, two years of college-level mathematics, and have gained enough mathematical maturity to be able to read and write proofs. It does not assume the usual background in algebra, set theory, and topology that is required by more advanced texts. It does attempt to guide readers to some of the deeper aspects of the subject, and in particular to some of the important interconnections with topology. Those parts of algebra and topology that are needed to understand the presentation are developed within the text itself. There is a separate appendix that covers the basic notions, notations, and theorems from set theory that are occasionally needed.
Boolean Rings
Boolean Algebras
Boolean Algebras Versus Rings
The Principle of Duality
Fields of Sets
Elementary Relations
Order
Infinite Operations
Topology
Regular Open Sets
Subalgebras
Homomorphisms
Extensions of Homomorphisms
Atoms
Finite Boolean Algebras
Atomless Boolean Algebras
Congruences and Quotients
Ideals and Filters
Lattices of Ideals
Maximal Ideals
Homomorphism and Isomorphism Theorems
The Representation Theorem
Canonical Extensions
Complete Homomorphisms and Complete Ideals
Completions
Products of Algebras
Isomorphisms of Factors
Free Algebras
Boolean σ-algebras
The Countable Chain Condition
Measure Algebras
Boolean Spaces
Continuous Functions
Boolean Algebras and Boolean Spaces
Duality for Ideals
Duality for Homomorphisms
Duality for Subalgebras
Duality for Completeness
Boolean σ-spaces
The Representation of σ-algebras
Boolean Measure Spaces
Incomplete Algebras
Duality for Products
Sums of Algebras
Isomorphisms of Countable Factors
Epilogue
A Set Theory
B Hints to Selected Exercises