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Introduction to Boolean Algebras

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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 p2 = 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.


Date Received: 
Monday, December 29, 2008
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Steven Givant and Paul Halmos
Undergraduate Texts in Mathematics
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Leon Harkleroad
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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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Friday, July 24, 2009