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This book is an informal though systematic series of lectures on Boolean algebras. It contains background chapters on topology and continuous functions and includes hundreds of exercises as well as a solutions manual.
- Sales Rank: #1609212 in Books
- Published on: 2008-12-02
- Original language: English
- Number of items: 1
- Dimensions: 9.21" h x 1.25" w x 6.14" l, 2.10 pounds
- Binding: Hardcover
- 574 pages
Review
From the reviews:
“This is an excellent and much-needed comprehensive undergraduate textbook on Boolean algebras. It contains a complete and thorough introduction to the fundamental theory of Boolean algebras. Aimed at undergraduate mathematics students, the book is, in the first author’s words, “a substantially revised version of Paul Halmos’ “Lectures on Boolean Algebras.” It certainly achieves its stated goal of “steering a middle course between the elementary arithmetic aspects of the subject” and “the deeper mathematical aspects of the theory” of Boolean algebras.”
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“The book is written for undergraduate students who already have skills in proving theorems. However, since the proofs are so detailed and clear, it could work well as a text for a second or even first course involving substantial proofs. For this reason, it would also make a great book for a student doing independent study. The text is somewhat informal in the sense that sometimes proofs appear in the prose rather than under the heading, “Proof”, but it is always clear when this is being done. Though the book starts with an introduction to Boolean rings, knowledge of group theory or rings is not a prerequisite for using the book.”
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“In summary, “Introduction to Boolean algebras” is a gem of a text which fills a long-standing gap in the undergraduate literature. It combines the best of both worlds by rigorously covering all the fundamental theorems and topics of Boolean algebra while at the same time being easy to read, detailed, and well-paced for undergraduate students. It is my most highly recommended text for undergraduates studying Boolean algebras.”
(Natasha Dobrinen. The Bulletin of Symbolic Logic, Vol. 16 (2), June 2010: 281-282)
"Introduction to Boolean Algebras … is intended for advanced undergraduates. Givant (Mills College) and Halmos … using clear and precise prose, build the abstract theory of Boolean rings and algebras from scratch. … the necessary topological material is developed within the book and an appendix on set theory is included. … Includes an extensive bibliography and more than 800 exercises at all levels of difficulty. Summing Up: Highly recommended. Upper-division undergraduates, graduate students, researchers, and faculty." (S. J. Colley, Choice, Vol. 46 (10), June 2009)
“The authors have written a book for advanced undergraduates and beginning graduate students. … The authors start with the definition of Boolean rings and Boolean algebras, give examples and basic facts and compare both notions. … 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.” (Martin Weese, Zentralblatt MATH, Vol. 1168, 2009) From the Back Cover
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.
Most helpful customer reviews
18 of 20 people found the following review helpful.
Superb introduction. (But one lecture is somewhat confused.)
By Michael Hardy
As always, Halmos is an excellent expositor. The brief first chapter has been known to scare away physicists and other intelligent people who don't happen to know ring theory, but fortunately the first chapter can be skipped. The book is at the right level to be used by mathematics graduate students learning Boolean algebras and Stone spaces for the first time.
Some people draw a sharp distinction between the concepts of "Boolean space" (a totally disconnected compact Hausdorff space) and "Stone space", the difference being that a Stone space is the Stone space _of_ a Boolean algebra. A Boolean algebra's Stone space is the space of all of its 2-valued homomorphisms with the topology of pointwise convergence of nets of such homomorphisms. That every Boolean space is the Stone space of some Boolean algebra (namely, the Boolean algebra of all of its clopen subsets) is one of the important facts of "Stone's duality". Halmos never mentions the phrase "Stone space", but he proves the basic facts about "Stone's duality": that the category of Boolean algebras and Boolean homomorphisms is the opposite of the category of Boolean spaces and continuous functions.
The one lecture that is somewhat confused is #21. I wrote earlier "Much of that section is founded upon an error -- that a Boolean algebra may have various non-isomorphic completions, of which one is 'minimal'". Someone has pointed out that Halmos does not actually say that any non-minimal completions actually exist, and that he proves a proposition at the end of Lecture 21 that entails that they do not. I doubt that many reasonable readers would think that Halmos didn't think that are some non-minimal completions. On page 93 he writes that all minimal completions are isomorphic to each other, but he never gets around to saying that all completions are isomorphic. Then, on page 96, in exercise 4, he says "Prove that B and C are in a natural sense completions of A", where A, B, and C have just been defined. The two Boolean algebras B and C are quotient algebras that Halmos showed on pages 67--68 to be non-isomorphic. Does this mean he believes non-isomorphic completions exist? Or only that the "natural sense" in which B and C are both completions of A is not the same sense of "completion" defined in the foregoing lecture? The latter is possible, but one can only wish Halmos had said so.
One other minor imperfection irritates me: Halmos uses the word "non-atomic" rather than the much better term "atomless". The problem with "non-atomic" is that it may be mistaken for "not atomic", and that is a quite different thing. And he first says that a certain Boolean algebra is non-atomic one page before he defines that concept.
4 of 7 people found the following review helpful.
ONE OF THE BEST
By Avigail
A very good book, going through the stuff in a very nice kind. Giving you intuitive insights, easy proofs, and written with a large sense of humor.
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