Sheaves of Algebras over Boolean Spaces
Sheaves of Algebras over Boolean Spaces comprehensively covers sheaf theory as applied to universal algebra. Sheaves decompose general algebras into simpler pieces called the stalks. A classical case is commutative von Neumann regular rings, whose stalks
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Sheaves of Algebras over Boolean Spaces
Arthur Knoebel 8412 Cherry Hills Drive, N.E. Albuquerque, NM 87111-1027 USA [email protected]
ISBN 978-0-8176-4218-1 e-ISBN 978-0-8176-4642-4 DOI 10.1007/978-0-8176-4642-4 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011940976 Mathematics Subject Classification (2010): 03C05, 06E15, 08-02, 08A05, 08A30, 08A40, 08B26, 18A22, 18F20 c Springer Science+Business Media, LLC 2012 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.birkhauser-science.com)
To the memory of ALFRED L. FOSTER, who set me to work representing algebras.
Preface
My involvement in the line of research leading to this book began in 1963 when I was a graduate student working under the direction of Alfred Foster, and was first learning about representing algebras as subdirect products. In particular, for a starter I learned that Stone’s representation theorem was valid not just for Boolean algebras but for any class of algebras satisfying the identities of a primal algebra. Foster perceived in these algebras a Boolean part whose representation theory could be levered into representing many other kinds of algebras. The broad motivation was to break up a complicated algebra into simpler pieces; if the pieces could be understood, then hopefully so could the whole algebra. The obvious decomposition to try first is a direct product. The advantage of direct products is the simplicity of their construction. The overwhelming disadvantage is that most algebras are indecomposable in this sense, and even when decomposable there may be no ultimate refinement. Subdirect products overcome both of these liabilities, as first demonstrated by Garrett Birkhoff. The main drawback to subdirect products is that, while factors may be commonplace and well understood, the transfer of an argument from the components to the whole algebra may fail because one may not know in sufficient detail how the components fit together to form the original algebra. Thus one grafts topological spaces onto subdirect products to form significantly superior sheaves. Elements of the subdirect product become continuous functions, and are easier to recognize. Boolean spaces are often used since they arise naturally in representing Boolean algebras and have been th
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