Introduction to Boolean Algebras / Undergraduate Texts in Mathematics (PDF)
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...
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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 isavailable for instructors who adopt the text for a course.
- Autoren: Steven Givant , Paul Halmos
- 2008, 2009, 574 Seiten, Englisch
- Verlag: Springer-Verlag GmbH
- ISBN-10: 0387684360
- ISBN-13: 9780387684369
- Erscheinungsdatum: 10.12.2008
Abhängig von Bildschirmgrösse und eingestellter Schriftgrösse kann die Seitenzahl auf Ihrem Lesegerät variieren.
- Dateiformat: PDF
- Grösse: 4.04 MB
- Ohne Kopierschutz
- Vorlesefunktion
“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 PaulHalmos’ “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
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