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Introduction to Boolean algebras. (English) Zbl 1168.06001

Undergraduate Texts in Mathematics. New York, NY: Springer (ISBN 978-0-387-40293-2/hbk; 978-0-387-68436-9/ebook). xiv, 574 p. (2009).
This book is a substantially revised version of the classic book [Lectures on Boolean algebras. Toronto etc.: Van Nostrand (1963; Zbl 0114.01603)] by P. Halmos. The authors have written a book for advanced undergraduates and beginning graduate students. The book does not assume knowledge of the usual background in algebra, set theory and topology. Those parts from algebra and topology which are needed are completely developed in the text. The basic facts from set theory that are necessary for the understanding are collected in an appendix.
The book is divided into 45 chapters. The first part of the book (up to Chapter 28) deals with the more basic facts, such as arithmetical and algebraic aspects of Boolean algebras. In the second part of the book, the authors deal with more advanced facts. Here they present the fundamental duality theorems for Boolean algebras and Boolean spaces.
The authors start with the definition of Boolean rings and Boolean algebras, give examples and basic facts and compare both notions. They go on with the principle of duality and introduce set algebras; they define the finite-cofinite as well as the countable-cocountable algebra and consider interval algebras. They investigate the connection of Boolean algebras with lattices and partial orders and deal with infinite operations. In the following two chapters (Chapters 9 and 10) they explain the connections between Boolean algebras and topological spaces and show the importance of regular open sets. They go on with subalgebras and homomorphism and prove the Sikorski extension theorem. The next three chapters deal with atoms, finite Boolean algebras and atomless Boolean algebras. The authors give a proof of the fact that any two countable atomless Boolean algebras are isomorphic. The next three chapters (Chapters 17–19) deal with congruences, quotients, ideals, filters and the lattice of ideals. They continue with maximal ideals, homomorphism and isomorphism theorems and show that each Boolean algebra is isomorphic to a field of sets. They introduce canonical extensions and show that each Boolean algebra has a unique canonical extension. In Chapters 24 and 25, the authors deal with complete homomorphisms and complete ideals and show the uniqueness theorem for completions. Then they investigate products of Boolean algebras and show that if two \(\sigma\)-complete Boolean algebras are factors of one another, then they are isomorphic. They present the Halmos example of two nonisomorphic Boolean algebras \(A\) and \(B\) with \(A^2 \cong B^2\). Chapter 28 deals with free algebras.
The second part is more advanced and is centered on the fundamental duality theorems for Boolean algebras and Boolean spaces. The authors start with Boolean \(\sigma\)-algebras and introduce some notions from topology. So they show the Baire category theorem. They go on with the countable chain condition and measure algebras. The following three chapters (Chapters 32–34) deal with Boolean spaces, continuous functions and the Stone representation theorem. The next four chapters deal with duality questions: duality for ideals, duality for homomorphisms, duality for subalgebras and duality for completeness. The authors continue with Boolean \(\sigma\)-spaces, the representation of \(\sigma\)-algebras and Boolean measure spaces. The following chapter deals with incomplete Boolean algebras. Here the authors investigate the quotients of Boolean algebras modulo an ideal. They go on with the duality for products and consider sums of algebras. In the last chapter they deal with a famous example of Halmos: There are two countable Boolean algebras \(A\) and \(B\) with \(A \cong A \times B \times B\) but \(A \not\cong A \times B\).
There are a large number of exercises of varying level of difficulty. Hints for the solutions of the harder problems are given in an appendix. A detailed solutions manual for all exercises is available for instructors.
The book can serve as a basis for a variety of courses. The authors give several hints for this in the preface.

MSC:

06-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordered structures
06E05 Structure theory of Boolean algebras
06E10 Chain conditions, complete algebras
06E15 Stone spaces (Boolean spaces) and related structures

Citations:

Zbl 0114.01603
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