Paul Halmos: Introduction to Boolean Algebras

Cover: 9781441923240 | Introduction to Boolean Algebras | Paul Halmos (u. a.) | Taschenbuch

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Beschreibung

The theory of Boolean algebras was created in 1847 by the English mat- matician George Boole. He conceived it as a calculus (or arithmetic) suitable for a mathematical analysis of logic. The form of his calculus was rather di?erent from the modern version, which came into being during the - riod 1864¿1895 through the contributions of William Stanley Jevons, Aug- tus De Morgan, Charles Sanders Peirce, and Ernst Schr¿ oder. A foundation of the calculus as an abstract algebraic discipline, axiomatized by a set of equations, and admitting many di?erent interpretations, was carried out by Edward Huntington in 1904. Only with the work of Marshall Stone and Alfred Tarski in the 1930s, however, did Boolean algebra free itself completely from the bonds of logic and become a modern mathematical discipline, with deep theorems and - portantconnections toseveral otherbranchesofmathematics, includingal- bra,analysis, logic, measuretheory, probability andstatistics, settheory, and topology. For instance, in logic, beyond its close connection to propositional logic, Boolean algebra has found applications in such diverse areas as the proof of the completeness theorem for ?rst-order logic, the proof of the Lo ¿ s conjecture for countable ?rst-order theories categorical in power, and proofs of the independence of the axiom of choice and the continuum hypothesis ? in set theory. In analysis, Stone¿s discoveries of the Stone¿Cech compac- ?cation and the Stone¿Weierstrass approximation theorem were intimately connected to his study of Boolean algebras.

Zusammenfassung

This book is an informal yet systematic presentation of lectures given by the author on Boolean algebras. The author's style is characteristically bold and fresh. He treats Boolean algebras, develops some intriguing ideas, and provides rare insights. Exercises are generously sprinkled through the text for course study. The second edition has been greatly expanded and rewritten, specific changes include:

* More detail explanations of the material in every section, making the text more accessible to undergraduates

* Three times as many exercises as well as a solutions manual

* A more careful explanation of the relationship between Boolean rings and Boolean algebras has been added;

* thirteen new chapters, including ones on topology and continuous functions and others on the extension theorem for homomorphisms, congruences and quotient algebras, lattice of ideals, and duality theory for products.

Inhaltsverzeichnis

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 s-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.- Duality for Completeness.- Boolean s-spaces.- The Representation of s-algebras.- Boolean Measure Spaces.- Incomplete Algebras.- Duality for Products.- Sums of Algebras.- Isomorphisms of Countable Factors.

Details

Erscheinungsjahr:	2010
Fachbereich:	Grundlagen
Genre:	Importe , Mathematik
Rubrik:	Naturwissenschaften & Technik
Medium:	Taschenbuch
Inhalt:	xiv 574 S. 10 s/w Illustr. 574 p. 10 illus.
ISBN-13:	9781441923240
ISBN-10:	1441923241
Sprache:	Englisch
Einband:	Kartoniert / Broschiert
Autor:	Halmos, Paul Givant, Steven
Auflage:	Softcover reprint of hardcover 1st edition 2009
Hersteller:	Springer US Springer New York
Verantwortliche Person für die EU:	Springer Verlag GmbH, Tiergartenstr. 17, D-69121 Heidelberg, juergen.hartmann@springer.com
Maße:	235 x 155 x 32 mm
Von/Mit:	Paul Halmos (u. a.)
Erscheinungsdatum:	19.11.2010
Gewicht:	0,879 kg