Goguen Categories A Categorical Approach to L-fuzzy Relations
Goguen categories extend the relational calculus and its categorical formalization to the fuzzy world. Starting from the fundamental concepts of sets, binary relations and lattices this book introduces several categorical formulations of an abstract theor
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TRENDS IN LOGIC Studia Logica Library VOLUME 25 Managing Editor Ryszard Wójcicki, Institute of Philosophy and Sociology, Polish Academy of Sciences, Warsaw, Poland Editors Vincent F. Hendricks, Department of Philosophy and Science Studies, Roskilde University, Denmark Daniele Mundici, Department of Mathematics “Ulisse Dini”, University of Florence, Italy Ewa Orłowska, National Institute of Telecommunications, Warsaw, Poland Krister Segerberg, Department of Philosophy, Uppsala University, Sweden Heinrich Wansing, Institute of Philosophy, Dresden University of Technology, Germany
SCOPE OF THE SERIES
Trends in Logic is a bookseries covering essentially the same area as the journal Studia Logica – that is, contemporary formal logic and its applications and relations to other disciplines. These include artificial intelligence, informatics, cognitive science, philosophy of science, and the philosophy of language. However, this list is not exhaustive, moreover, the range of applications, comparisons and sources of inspiration is open and evolves over time.
Volume Editor Ewa Orłowska
The titles published in this series are listed at the end of this volume.
GOGUEN CATEGORIES A Categorical Approach to L-fuzzy Relations
by
MICHAEL WINTER Brock University, St. Catharines, ON, Canada
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-1-4020-6163-9 (HB) ISBN 978-1-4020-6164-6 (e-book)
Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. www.springer.com
Printed on acid-free paper
All Rights Reserved © 2007 Springer No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work.
To my family
Contents
INTRODUCTION
ix
1. SETS, RELATIONS, AND FUNCTIONS
1
2. LATTICES 2.1 Galois correspondences and residuated operations 2.2 Distributive lattices 2.3 Brouwerian lattices 2.4 Boolean algebras 2.5 Special elements 2.6 Fixed points 2.7 The complete Brouwerian lattice of antimorphisms 2.8 Filters 2.9 Lattice-ordered semigroups
5 13 17 18 20 22 23 25 31 40
3. L-FUZZY RELATIONS 3.1 Basic operations and properties 3.2 Crispness 3.3 Operations derived from lattice-ordered semigroups
43 43 48 52
4. CATEGORIES OF RELATIONS 4.1 Categories 4.2 Allegories 4.3 Distributive allegories 4.4 Division allegories 4.5 Dedekind categories 4.6 Relational constructions in Dedekind categories 4.7 The Dedekind category of antimorphisms 4.8 Scalars and crispness in Dedekind categories 4.9 Schr¨ oder categories 4.10 Formal languages of relational categories
55 55 57 63 65 68 74 75 79 85 86
vii
viii
GOGUEN CATEGORIES
5. CATEGORIES OF L-FUZZY RELATIONS 5.1 Arrow categories 5.2 The arrow category of antimorphisms 5.3 Arrow categories w
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