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Publisher:

Springer

Publication Date:

2009

Number of Pages:

574

Format:

Hardcover

Series:

Undergraduate Texts in Mathematics

Price:

59.95

ISBN:

9780387402932

Category:

Textbook

The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by , on ]

Leon Harkleroad

06/8/2009

The combination of Paul Halmos and an exposition of Boolean algebras might ring a few bells. In fact, *Introduction to Boolean Algebras* is a decidedly expanded version of the now out-of-print *Lectures on Boolean Algebras*, one of many Halmos works in the MAA’s Basic Library List.

*Introduction* nearly quadruples the number of pages of *Lectures,* from 147 to 574. And a typical page of the new book contains about 40% more words than one in the old. Yet in many ways *Introduction* very much resembles *Lectures.* So what changes have been made, and why?

As its title suggests, the original (1963) version consisted of lecture notes for a course that Halmos taught at the University of Chicago. His preface did not specify the background of the students, but early on the text casually refers, without any further explanations, to the von Neumann definition of ordinal, the Klein four-group, and Lebesgue measurable sets and Borel sets. In contrast, *Introduction* explicitly “is aimed at undergraduates who have studied, say, two years of college-level mathematics, and … does not assume the usual background in algebra, set theory, and topology that is required by more advanced texts.”

Thus *Lectures* opens with the sentences “An element p of a ring is *idempotent* if p^{2} = p. A *Boolean ring* is a ring with unit in which every element is idempotent.” *Introduction* starts, “A ring is an abstract version of arithmetic,” and through the concrete examples of the integers and the integers mod 2, works up to the definitions of ring and Boolean ring. Similarly, *Introduction* offers a chapter on basic topology and an appendix on naive set theory (which owes a bit to Halmos’ book on the subject) to supply background that *Lectures* took for granted.

Besides providing such preliminary material, *Introduction* develops its core content in a more leisurely fashion. Results receive more detailed proofs, and many of the laconic assertions in *Lectures* get fleshed out here. In broad terms, the two books cover about the same territory. This ranges from the very basics to atomic Boolean algebras to the countable chain condition to Boolean spaces and their duality with Boolean algebras to Hanf’s construction of Boolean algebras A and B such that A is isomorphic to AxBxB but not isomorphic to AxB.

The final three chapters of *Lectures*, covering retracts and projective and injective algebras, did not make their way into the new version. Otherwise, most of the original material can be found in *Introduction,* although the presentation has been significantly reorganized. Moreover, various topics and results, absent from *Lectures,* show up — canonical extensions and isomorphism of countable atomless Boolean algebras via back-and-forth argument, to name a couple.

*Introduction* incorporates not only most of the topics and results from *Lectures,* but much of the text as well. In particular, the “expressions of personal opinion and irreverent viewpoint,” to which Halmos referred in the preface to *Lectures,* remain. Halmos’ distinctive voice certainly did not get lost in the rewrite. (The preface to *Introduction* notes, “This revision of Halmos’s book was planned and initially executed by both authors. Due to declining health, however, Halmos was not able to review the later versions of the manuscript.”)

The exercises underwent even more expansion than the text proper, from 139 to 706, some of them routine, others real challenges. Often a proof in the text invokes a result from an exercise in a previous chapter. And, unlike the original, *Introduction* contains an appendix of “hints” to many of the exercises. These vary from terse to highly detailed, with one lasting over three pages.

Another new feature of *Introduction* is a bibliography of 85 items. The book also sports a somewhat idiosyncratically organized, 33-page index.

Will *Introduction* fill the spot on the Basic Library List vacated by its forerunner? No doubt, *Lectures* earned that spot, in part, by the concise elegance of its presentation. However, the revision very helpfully makes the material accessible to a wider audience, while still preserving much of what made the original a classic.

Leon Harkleroad has contributed MAA Reviews of books on a variety of topics, but this is the first time he has written a review full of Boole.

Preface.- 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 o-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.- Dualtiy for Completeness.- Boolean o-spaces.- The Representation of o-algebras.- Boolean Measure Spaces.- Incomplete Algebras.- Dualtiy for Products.- Sums of Algebras.- Isomorphisms of Countable Factors.-Epilogue.- Set Theory.- Hints to Selected Exercises.- References.- Index

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