Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory

721 Pages·1999·22.068 MB·English

by  Ulrich Höhle, Stephen Ernest Rodabaugh (auth.), Ulrich Höhle, Stephen Ernest Rodabaugh (eds.)

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, photo
copying, 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 com
pactifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 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 with
out 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 inclu
sion 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 circumstan
cial 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