Mathematics of Fuzzy Sets Logic, Topology, and Measure Theory
Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory is a major attempt to provide much-needed coherence for the mathematics of fuzzy sets. Much of this book is new material required to standardize this mathematics, making this volume a reference
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THE HANDBOOKS OF FUZZY SETS SERIES Series Editors Didier Dubois and Henri Prade IRIT, Universite Paul Sabotier, Toulouse, France
FUNDAMENTALS OF FUZZY SETS, edited by Didier Dubois and Henri Prade MATHEMATICS OF FUZZY SETS: Logic, Topology, and Measure Theory, edited by Ulrich Höhle and Stephen Ernest Rodabaugh FUZZY SETS IN APPROXIMATE REASONING AND INFORMATION SYSTEMS, edited by James C. Bezdek, Didier Dubois and Henri Prade FUZZY MODELS AND ALGORITHMS FOR PATTERN RECOGNITION AND IMAGE PROCESSING, by James C. Bezdek, James Keller, Raghu Krisnapuram andNikhil R. Pal FUZZY SETS IN DECISION ANALYSIS, OPERATIONS RESEARCH AND STATISTICS, edited by Roman Slowinski FUZZY SYSTEMS: Modeling and Control, edited by Hung T. Nguyen and Michio Sugeno PRACTICAL APPLICATIONS OF FUZZY TECHNOLOGIES, edited by HansJürgen Zimmermann
MATHEMATICS OF FUZZY SETS LOGIC, TOPOLOGY, AND MEASURE THEORY
edited by
ULRICH HÖHLE Fachbereich Mathematik Bergische Universität, Wuppertal, Germany and
STEPHEN ERNEST RODABAUGH Department of Mathematics and Statistics Youngstown State University, Youngstown, Ohio, USA
Springer Science+Business Media, LLC
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Library of Congress Cataloging-in-Publication Data Mathematics offuzzy sets : logic, topology, and measure theory / edited by Ulrich Rohle and Stephen Emest Rodabaugh. p. cm. -- (The handbooks offuzzy sets series ; 3) Includes bibliographical references and index. ISBN 978-1-4613-7310-0
ISBN 978-1-4615-5079-2 (eBook)
DOI 10.1007/978-1-4615-5079-2
1. Fuzzy sets. 2. Fuzzy mathematics. 1. Rohle, Ulrich. II. Rodabaugh, Stephen Emest. III. Series: Randbooks of Fuzzy Sets series ; FSRS 3. QA248.5.M37 1999 511.3'22--dc21 98-45584 CIP Copyright © 1999 by Springer Science+Business Media New York Originally published by Kluwer Academic Publishers in 1999 AII rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, mechanical, photocopying, recording, or otherwise, without the prior written permis sion of the publisher, Springer Science+Business Media, LLC Printed on acid-free paper.
CONTENTS Authors and Editors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Foreword . . . . . .
. ix
Introduction . . . .
1
1. Many-valued logic and fuzzy set theory. . . . . . . . . . . . . . . . . .. S. Gottwald
5
2. Powerset operator foundations for poslat fuzzy set theories and topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 S.E. Rodabaugh Introductory notes to Chapter 3 . . . . . . . . . . . . . . . U. Hohle
117
3. Axiomatic foundations of fixed-basis fuzzy topology. . .. U. Hohle and A.P. Sostak
123
4. Categorical foundations of variable-basis fuzzy topology. . . . S.E. Rodabaugh
273
5. Characterization of L-topologies by L-valued neighborhoods. . . . . . 389 U. Hohle 6. Separation axioms: Extension of mappings and embedding of spaces .. 433 T. Kubiak 7. Separation axioms: Representation theorems, compactness, and compactif
