MATHEMATICS OF FUZZY SETS

LOGIC, TOPOLOGY, AND MEASURE

THEORY

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 Hans-Jürgen 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

.... " Electronic Services <http://www.wkap.nl>

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. . . . . . . . . . . . . . . . . .. 5 S. Gottwald

2. Powerset operator foundations for poslat fuzzy set theories and topo-logies ........................................ 91

S.E. Rodabaugh

Introductory notes to Chapter 3 . . . . . . . . . . . . . . . 117 U. Hohle

3. Axiomatic foundations of fixed-basis fuzzy topology. . .. 123 U. Hohle and A.P. Sostak

4. Categorical foundations of variable-basis fuzzy topology. . . . 273 S.E. Rodabaugh

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 com­pactifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 481

S.E. Rodabaugh

8. Uniform spaces ............. . 553 W. Kotze

9. Extensions of uniform space notions ................. " 581 M.H. Burton and J. Gutierrez Garcia

10. Fuzzy real lines and dual real lines as poslat topological, uniform, and metric ordered semirings with unity . . . . . . . . . . . . . . .. . 607 S.E. Rodabaugh

11. Fundamentals of generalized measure theory ................ 633 E.P. Klement and S. Weber

12. On conditioning operators ................ . . .. 653 U. Hohle and S. Weber

13. Applications of decomposable measures. . .675 E. Pap

14. Fuzzy random variables revisited .. . . 701 D.A. Ralescu

Index ........................................... 711

Authors and Editors

Michael Howard Burton Department of Mathematics Rhodes University Grahamstown, 6140, South Africa.

Javier Gutierrez Garda Matematika Saila Euskal Herriko Unibertsitatea E-48080 Bilbo, Spain.

Siegfried Gottwald Institut fUr Logik und Wissenschaftstheorie Universitat Leipzig Augustusplatz 9 D-04109 Leipzig, Germany.

Ulrich Hohle Fachbereich 7 Mathematik Bergische Universiat Wuppertal GauBstraBe 20 D-42097 Wuppertal, Germany.

Erich Peter Klement Institut fur Mathematik Johannes Kepler Unversitat Linz A-4040 Linz, Austria.

Wesley Kotze Depatment of Mathematics Rhodes University Grahamstown, 6140, South Africa.

Tomasz Kubiak Wydzial Matematyki i Informatyki ul. Matejki 48/49 PL-60-769 Poznan, Poland.

Endre Pap Institute of Mathematics 21000 Novi Sad Trg Dositeja Obradovica 4 Yugoslavia.

Dan A. Ralescu Department of Mathematical Sciences University of Cincinnati Cincinnati, Ohio 45221-0025, USA.

Stephen Ernest Rodabaugh Department of Mathematics and Statistics Youngstown State University. Youngstown, Ohio 44555-3609, USA.

Alexander P. Sostak Department of Mathematics University of Latvia LV-1586 Riga, Latvia.

Siegfried Weber Fachbereich Mathematik Johannes Gutenberg-Universitat Mainz 0-55099 Mainz, Germany.

Foreword

This book, Volume 3 of the The Handbooks of Fuzzy Sets Series, is intended to serve as a reference work for certain fundamental mathematical aspects of fuzzy sets, including mathematical logic, measure and probability theory, and especially general topology.

Its place in this Handbook series has a two-fold historical root: the need for standardization, both conceptually and notationally, in the mathematics of fuzzy sets; and the prominent role in the development of the mathematics of fuzzy sets played by the International Seminar on Fuzzy Set Theory-also known as the Linz Seminar-held annually since 1979 in Linz, Austria.

By the late 1980's, it was apparent to a broad spectrum of workers in fuzzy sets that the utility of the mathematics being rapidly developed in fuzzy sets was significantly restricted in every way by the overall incoherence of its literature, both conceptually and notationally. In keeping with the long-standing traditions of the Linz Seminar, many Seminar participants felt this problem should be addressed by a reference work which would standardize definitions, notations, and concepts, as well as make the mathematics of fuzzy sets more accessible and usable for the world-wide scientific community. This volume, in significant measure the work of Seminar participants, is an attempt to be such a reference work.

Given the role which this volume was intended to play, the editors kept always in mind the two critical criteria of a reference work: standardization and state-of-the-art. These criteria were often intertwined: in many of the chapters contained herein, significant new developments were sometimes necessary to facilitate standardization. It is the editors' hope that this volume so satisfies these two criteria that readers and workers from many backgrounds and interests will find it a useful reference tool as well as a platform for future research.

No work of this purpose and scope can hope for any degree of success with­out the support of many talented and committed individuals. The editors are grateful to the editors of the Handbook series, D. Dubois and H. Prade, for the opportunity to organize and edit this volume and for their supporting the inclu­sion of the mathematics of fuzzy sets in the Handbook series. Deepest thanks go to the chapter authors for their willingness to be a part of this important project and for their willingness to write, rewrite, rewrite yet again, polish, and enlarge their chapters to collectively produce this reference work. Appreciation is also expressed to the editors' respective universities for the use of facilities and secretarial and system staff help. Finally, gratitude is expressed to Mr. Gary Folven of Kluwer Academic Publishers for his cooperation, support, and infinite patience.

The Editors

Series Foreword

Fuzzy sets were introduced in 1965 by Lotfi Zadeh with a view to reconcile mathematical modeling and human knowledge in the engineering sciences. Since then, a considerable body of literature has blossomed around the concept of fuzzy sets in an incredibly wide range of areas, from mathematics and logics to traditional and advanced engineering methodologies (from civil engineering to computational intelligence). Applications are found in many contexts, from medicine to finance, from human factors to consumer products, from vehicle control to computational linguistics, and so on. Fuzzy logic is now currently used in the industrial practice of advanced information technology.

As a consequence of this trend, the number of conferences and publications on fuzzy logic has grown exponentially, and it becomes very difficult for students, newcomers, and even scientists already familiar with some aspects of fuzzy sets, to find their way in the maze of fuzzy papers. Notwithstanding circumstan­cial edited volumes, numerous fuzzy books have appeared, but, if we except very few comprehensive balanced textbooks, they are either very specialized monographs, or remain at a rather superficial level. Some are even misleading, conveying more ideology and unsustained claims than actual scientific contents.

What is missing is an organized set of detailed guidebooks to the relevant literature, that help the students and the newcoming scientist, having some preliminary knowledge of fuzzy sets, get deeper in the field without wasting time, by being guided right away in the heart of the literature relevant for her or his purpose. The ambition of the HANDBOOKS OF FUZZY SETS is to address this need. It will offer, in the compass of several volumes, a full picture of the current state of the art, in terms of the basic concepts, the mathematical developments, and the engineering methodologies that exploit the concept of fuzzy sets.

This collection will propose a series of volumes that aim at becoming a useful source of reference for all those, from graduate students to senior researchers, from pure mathematicians to industrial information engineers as well as life, human and social sciences scholars, interested in or working with fuzzy sets. The original feature of these volumes is that each chapter is written by one or

several experts in the concerned topic. It provides introduction to the topic, outlines its development, presents the major results, and supplies an extensive bibliography for further reading.

The core set of volumes are respectively devoted to fundamentals of fuzzy set, mathematics of fuzzy sets, approximate reasoning and information systems, fuzzy models for pattern recognition and image processing, fuzzy sets in decision analysis, operations research and statistics, fuzzy systems modeling and control, and a guide to practical applications of fuzzy technologies.

Didier DUBOIS Henri PRADE Toulouse