Texture Spaces (Studies in Fuzziness and Soft Computing, 411) 3031397479, 9783031397479

Table of contents :
Preface
References
Acknowledgements
Contents
1 Introduction
1.1 Lattices
1.2 Completely Distributive Lattices
References
2 Textures
2.1 Basic Concepts and Results
2.2 p-Set and q-Set
2.3 Equivalences of Completely Distributivity in Texture Spaces
2.4 Simple and Plain Textures
References
3 Brown's Representation Theorem
3.1 Textural Isomorphisms
3.2 Product of Textures
3.3 L-Fuzzy Set Texturing mathcalP(U)otimesmathcalML
3.4 Orthopair (Intuitionistic) Textures
References
4 Direlations
4.1 Basic Concepts
4.2 Textural Image of a Set: Sections
4.3 Composition of Direlations
4.4 Seriality and Injectivity
5 Rough Sets
5.1 Pawlak's Approximation Spaces
5.2 Generalized Approximation Spaces
5.3 Generalized Approximation Spaces with Two Domains of Discourse
5.4 Textural Rough Sets
5.5 Textural Definability
References
6 Basic Results in Rough Set Theory via Textures
6.1 Generalized Approximation Spaces and Discrete Textures
6.2 Definability in Generalized Approximation Spaces with Two Domains of Discourse
6.3 Revised Textural Rough Sets
6.4 Revised Rough Set Approximations
References
7 Fuzzy Rough Sets
7.1 Fuzzy Logical Connectives
7.2 Continuity of Fuzzy Logical Connectives
7.3 Textural Fuzzy Direlations
7.4 Fuzzy Direlations Defined by Fuzzy Logic Connectives
7.5 Adjointness and Duality
7.6 The Well-Known Fuzzy Rough Set Models Obtained by t-Fuzzy Direlations
7.7 Basic Properties of Fuzzy Relations
7.8 t-Fuzzy Rough Set Approximations with Two Domains of Discourse
7.9 Definability in t-Fuzzy Approximation Spaces
7.10 Basic Properties of Fuzzy Rough Sets with Two Domains of Discourse
7.11 Definability in Terms of Fuzzy Logic Connectives
7.12 Revised t-Fuzzy Rough Set Models
7.13 Revised Fuzzy Rough Set Models with Two Domains of Discourse
References
Appendix Index
Index

Citation preview

Studies in Fuzziness and Soft Computing

Murat Diker

Texture Spaces

Studies in Fuzziness and Soft Computing Volume 411

Series Editor Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

The series “Studies in Fuzziness and Soft Computing” contains publications on various topics in the area of soft computing, which include fuzzy sets, rough sets, neural networks, evolutionary computation, probabilistic and evidential reasoning, multi-valued logic, and related fields. The publications within “Studies in Fuzziness and Soft Computing” are primarily monographs and edited volumes. They cover significant recent developments in the field, both of a foundational and applicable character. An important feature of the series is its short publication time and world-wide distribution. This permits a rapid and broad dissemination of research results. Indexed by SCOPUS, DBLP, WTI Frankfurt eG, zbMATH, SCImago. All books published in the series are submitted for consideration in Web of Science.

Murat Diker

Texture Spaces

Murat Diker Department of Mathematics Hacettepe University Ankara, Türkiye

ISSN 1434-9922 ISSN 1860-0808 (electronic) Studies in Fuzziness and Soft Computing ISBN 978-3-031-39747-9 ISBN 978-3-031-39748-6 (eBook) https://doi.org/10.1007/978-3-031-39748-6 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To my wife Nilgün and our grandson Bora with love.

Preface

This book contains the texture theory and its relations to rough sets and fuzzy sets. It provides the intelligibility of the notions due to textures and presents concrete examples with easy explanations for the basic connections among the rough sets, fuzzy sets and textures, and the recent investigations. Many observations related to textures are scattered in the literature, and this book aims to combine the basic tools and theoretical arguments of textures in this respect. Prospective readers are not only mathematicians who interest in purely mathematical theories related to textures, but also engineers of information sciences who need theoretical models for their interdisciplinary studies due to rough sets and fuzzy sets. Therefore, I tried to make self-contained and reader friendly this source as much as possible. The theory of texture spaces was produced by Lawrence M. Brown and presented first by him at the 2nd International Conference of the Balcanic Union for Fuzzy Systems and Artificial Intelligence in 1992. The first formal paper about textures was published in the Special Issue of the Journal of Fuzzy Sets and Systems entitled “Topics of the Mathematics of Fuzzy Objects” by L. M. Brown and M. Diker in 1998 [1]. A texture is a family of subsets of a domain of discourse which is an alternative point-set-based setting for fuzzy sets. The notions of approximation operator and definable set of rough set theory are also key concepts in the theory of texture spaces. Recent investigations show that the theory of texture is one of the remarkable mathematical models for fuzzy sets and rough sets. However, the peculiar structure of textures makes difficult to understand the basic concepts and results of the theory of textures for the researchers who study real-life applications using fuzzy sets and rough sets. Texture space, as one of the main subjects of this book, is essentially motivated by fuzzy sets. Therefore, most of the basic arguments of fuzzy sets can be considered in terms of textures. Rough set theory whose creator is Zdzislaw Pawlak [2] arosed from the need for the solutions of the problems related to imperfect data with respect to information systems. The theory of fuzzy sets was created by Lotfi A. Zadeh [6], and it is also subjected to imperfect or imprecise data. However, the rough set theory uses indiscernibility relations while fuzzy set theory considers the membership degrees of the objects. It is remarkable to say that both of the theories are complementary to each other as it was discussed in detail by Pawlak and Skowron in vii

viii

Preface

[3]. Texture, as a purely mathematical structure, combines fuzzy set theory and rough set theory in various directions. I believe that this book will be a useful source for the researchers (computer scientists, engineers, mathematicians) who need development of the theoretical background for their works due to information sciences. The contents of the book consist of 7 chapters. Chapter 1 is devoted to some basic facts from lattice theory. Complete distributivity plays an important role in texture spaces and hence, we give a characterization for the complete distributivity in textures using the Raney’s theorem facilitating the proofs with respect to complete distributivity. In Chap. 2, Sect. 2.1 starts with one of the canonic examples of fuzzy lattices providing the motivation of textures. Section 2.2 presents the main tools of a texture called a p-set and q-set assuring a strong duality in almost all textural results. Section 2.3 focuses on the equivalences of completely distributivity, in particular in terms of p-set and q-set. Chapter 2 is ended with the important subclasses of textures called simple and plain textures. Chapter 3 presents Brown’s Representation Theorem for textures. In Sects. 3.1 and 3.2, textural isomorphism and the product of textures are discussed. Sections 3.3 and 3.4 discuss two important examples of textures called L-fuzzy texture and orthopair texture, respectively. In Chap. 4, the concept of a direlation is defined as a morphism between textures. A direlation is a generalization of the ordinary relation between sets, and it is compatible with the product of textures. The notions of image and preimage due to a direlation called section and presection, respectively, are the remarkable generalizations of Pawlak approximations. This unexpected connection makes these concepts even more valuable and so, in Sects. 4.2–4.4, together with seriality and injectivity classes, almost all properties of direlations are given in detail. A summary about Pawlak’s rough sets and generalizations presented by Yiyu Yao [4, 5] are given in Sects. 5.1, 5.2 and 5.3 taking into account the theoretical development of approximation spaces, respectively. Sections 5.4 and 5.5 discuss the textural rough sets and the connections to Pawlak’s rough set approximations and generalizations with a single or two domains of discourse. A further discussion is also devoted to textural definability in Sect. 5.5. Section 6.1 considers the formal connection between textures and rough sets. Textural definability will be used in Sect. 6.2 to prove the results due to definability in generalized approximation spaces. Chapter 6 is essentially devoted to the basic properties of rough set theory. In Sect. 6.3, the revised rough set approximations are improved using the textural arguments given in Sect. 6.4. Finally, Chap. 7 presents a detailed study on fuzzy rough sets. The combinations of fuzzy sets and rough sets can be realized using fuzzy logical connectives. Therefore, Sect. 7.1 reviews the information about the fuzzy logical connectives. The continuity of fuzzy logical connectives is widely used in the literature. It is known that some algebraic properties such as adjointness can be obtained from continuity condition. However, in most cases, continuity is also a necessary and significant condition for the basic results of fuzzy rough set approximations. Therefore, for the sake of the readers,

Preface

ix

Sect. 7.2 provides some basic topological arguments due to continuity of the binary relations defined on the cartesian product of the unit interval. The fuzzy version of a direlation, called t-fuzzy direlation, is a key concept for a general approach to fuzzy rough set models and hence, textural fuzzy direlations are discussed in Sect. 7.3. Section 7.4 argues the propositional fuzzy logic construction for a bridge between fuzzy direlations and fuzzy relations. Sections 7.5 and 7.6 are devoted to adjointness, duality and the well-known fuzzy rough set models. This makes it possible to express the concepts of seriality, injectivity, symmetricity and transitivity of a fuzzy relation in terms of continuous t-norms and implicators. These arguments can be found in Sect. 7.7. Further, Sects. 7.8 and 7.9 are devoted to t-fuzzy rough set approximations and definability. The results given here are used for the basic arguments due to fuzzy rough set approximations with two domains of discourse in Sects. 7.10 and 7.11. Finally, Sects. 7.12 and 7.13 argue the revised fuzzy rough set approximations. Ankara, Türkiye May 2023

Murat Diker

References 1. L.M. Brown, M. Diker, Ditopological texture spaces and intuitionistic sets. Fuzzy Sets Syst. 98, 217224 (1998) 2. Z. Pawlak, Rough Sets. Int. J. Comput. Info. Sci. 341–356 (1982) ˘ S27. 3. Z. Pawlak, A. Skowron. Rudiments of rough sets. Info. Sci. 177 (2007) 3âA ¸ 4. Y. Y. Yao, Relational interpretations of neigborhood operators and rough set approximations operators. Inf. Sci. 111(1-4), 239–259 (1998) 5. Y.Y. Yao, Combination of rough and fuzzy sets based on alpha-level sets, in Rough Sets and Data Mining: Analysis for Imprecise Data, eds. by T.Y. Lin, N. Cercone (Kluwer Academic Publishers, Boston, 1997) pp. 301–321 6. L.A. Zadeh. Inf. Control. 8(3), 338–353 (1965)

Acknowledgements

I would like to express my gratitude to Professor Janusz Kacprzyk, the Editor inChief of the book series, for his valuable encouragement. I am pleased to extend my appreciation to Professors Ay¸segül Altay U˘gur and Sadık Bayhan, who thoroughly reviewed the book and provided significant recommendations and feedback. I am also grateful to production staffs of Springer Nature, especially to Jayarani Premkumar and Femina Joshi A. I am indebted to journals of Fuzzy Sets and Systems, Information Sciences and International Journal of Approximate Reasoning.

xi

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Completely Distributive Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 4 8

2 Textures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Basic Concepts and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 p-Set and q-Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Equivalences of Completely Distributivity in Texture Spaces . . . . . 2.4 Simple and Plain Textures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 9 12 14 18 19

3 Brown’s Representation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Textural Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Product of Textures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 L-Fuzzy Set Texturing P(U ) ⊗ M L . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Orthopair (Intuitionistic) Textures . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21 21 27 37 39 45

4 Direlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Textural Image of a Set: Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Composition of Direlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Seriality and Injectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47 47 57 73 77

5 Rough Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.1 Pawlak’s Approximation Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.2 Generalized Approximation Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.3 Generalized Approximation Spaces with Two Domains of Discourse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.4 Textural Rough Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.5 Textural Definability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 xiii

xiv

Contents

6 Basic Results in Rough Set Theory via Textures . . . . . . . . . . . . . . . . . . . 6.1 Generalized Approximation Spaces and Discrete Textures . . . . . . . 6.2 Definability in Generalized Approximation Spaces with Two Domains of Discourse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Revised Textural Rough Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Revised Rough Set Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Fuzzy Rough Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Fuzzy Logical Connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Continuity of Fuzzy Logical Connectives . . . . . . . . . . . . . . . . . . . . . 7.3 Textural Fuzzy Direlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Fuzzy Direlations Defined by Fuzzy Logic Connectives . . . . . . . . . 7.5 Adjointness and Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 The Well-Known Fuzzy Rough Set Models Obtained by t-Fuzzy Direlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Basic Properties of Fuzzy Relations . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8 t-Fuzzy Rough Set Approximations with Two Domains of Discourse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.9 Definability in t-Fuzzy Approximation Spaces . . . . . . . . . . . . . . . . . 7.10 Basic Properties of Fuzzy Rough Sets with Two Domains of Discourse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.11 Definability in Terms of Fuzzy Logic Connectives . . . . . . . . . . . . . 7.12 Revised t-Fuzzy Rough Set Models . . . . . . . . . . . . . . . . . . . . . . . . . . 7.13 Revised Fuzzy Rough Set Models with Two Domains of Discourse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

107 107 113 120 125 137 139 139 145 156 179 187 190 193 202 211 215 223 227 231 245

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

Chapter 1

Introduction

A Texturing is a lattice of ordinary sets satisfying certain conditions which are compatible with the structure of a lattice of fuzzy subsets. In this chapter, we give some basic properties of lattices which are related to textures. For further information about lattices, we refer to [1].

1.1 Lattices As is known that the notion of (complete) distributivity is an algebraic property providing remarkable representations of lattices in terms of sets of a family. In other words, distributivity gives more concrete information about the structure of a lattice through the sets and the basic set operations. For instance, Birkhoff’s theorem says that any finite distributive lattice can be represented with the lattice of lower sets of the partial order of the join-irreducible elements [1]. A complete rings of sets is a family which is closed under intersections and unions. Raney showed that a complete lattice is completely distributive if and only if it is homeomorphic to a complete ring of sets [2]. The unit interval [0, 1] is a complete and completely distributive lattice with the ordinary order “≤” of real numbers. Moreover, the family of all fuzzy subsets of a given domain of discourse U , that is, the family F (U ) = { f | f : U → [0, 1]} is also a complete and completely distributive lattice where f denotes a function from U to [0, 1]. A basic motivation of a definition of a texture notion is to find an alternative point-set-based setting for such lattices. A texturing is a complete lattice of a family of some ordinary sets with respect to set inclusion satisfying the completely distributivity. From this point of view, Brown’s representation theorem provides a family of some crisp subsets of U as a point-set-based setting for F (U ) which will be called a texturing. Further, (complete) distributivity not only supply with a kind of ring of sets for F (U ), but also allows to perform operations in the classical sense © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Diker, Texture Spaces, Studies in Fuzziness and Soft Computing 411, https://doi.org/10.1007/978-3-031-39748-6_1

1

2

1 Introduction

related to points and sets in view of Raney’s theorem [3] given in the sequel (see, p-set and q-set). Now, let us start with a non-empty set L. A partially order ≤ is a binary relation on L satisfying the following conditions: For all r, s, t ∈ L, r ≤ r (reflexivity), r ≤ s and s ≤ r =⇒ r = s (anti-symmetry) and r ≤ s and s ≤ t =⇒ r ≤ t (transitivity). Then we call the pair (L , ≤) a partially ordered set. Let A ⊆ L with A = ∅. Then r ∈ L is called (i) a lower bound of A if r ≤ s for all s ∈ A, and (ii) an upper bound of A if s ≤ r for all s ∈ A, (iii) the greatest lower bound of A, if r is a lower bound of A, and s ≤ r for all lower bound s of A, (iv) the least upper bound of A if r is an upper bound of A, and r ≤ s for all upper bound  s of A. In this case, we write r = A or r = A, for the greatest lower bound or least upper bound of A, respectively. A partially ordered set (L , ≤) is called a lattice if for any two elements r, s ∈ L, the set {r, s} has the least upper bound and the greatest lower bound. Note that a mapping φ between two lattices L and L is a lattice isomorphism if it is a one-to-one and onto mapping preserving arbitrary meets and joins, that is, φ



     rj = φ(r j ) and φ rj = φ(r j )

j∈J

j∈J

j∈J

j∈J

for all {r j | j ∈ J } ⊆ L. A lattice L is said to be complete if for each non-empty subset A of L,  the least upper bound A (join), and the greatest lower bound A (meet) exist. If a lattice L has smallest and largest elements denoted by 0 and 1 respectively, then it is called a bounded lattice. The partially ordered set (P(U ), ⊆) is clearly a complete and bounded lattice with  j∈J

Aj =

 j∈J

Aj,



Aj =

j∈J



Aj

j∈J

where {A j | j ∈ J } ⊆ P(U ), 0 = ∅ and 1 = U . The following result will be useful in the sequel: Theorem 1.1.1 Let U be set and A ⊆ P(U ) where P(U ) is the family of all subsets of U . If U ∈ A and A is closed under arbitrary intersections, then (A, ⊆) is a complete lattice.

1.1 Lattices

3

Proof Clearly, (A, ⊆) is a partially ordered set. Let B ⊆ A and B = ∅. Note that

B ∈ P(U ) and by the assumption, we also have that B ∈ A. Hence B is the greatest lower bound of B in A. Let us consider the family C=



{A ∈ A |



B ⊂ A}.

Since C ∈ A and U ∈ A, C = ∅. Note that for all A ∈ B, A ⊆ C, that is C is an upper bound for B. Moreover, C is the least upper bound of B. Let L be a lattice and m ∈ L with m = 0. Then m is called (i) a molecule or join prime, if ∀r, s ∈ L , m ≤ r ∨ s ⇒ m ≤ r or m ≤ s, (ii) join irreducible, if ∀r, s ∈ L , m = r ∨ s ⇒ m = r or m = s. Clearly, if m is a molecule, then it is also join irreducible. Indeed, if m = r ∨ s, then m ≤ r ∨ s and so m ≤ r or m ≤ s. However, r ≤ m and s ≤ m yields that m = r and m = s. A lattice L is called distributive if for all r, s, t ∈ L, the equalities r ∧ (s ∨ t) = (r ∧ s) ∨ (r ∧ t), and r ∨ (s ∧ t) = (r ∨ s) ∧ (r ∨ t) hold. If L is distributive, then the notions of molecule and join-irreducible element are coincide [4]. That is, if m is join irreducible, then it is also a molecule. Let m ≤ r ∨ s. Then by distributivity, we have m = (r ∨ s) ∧ m = (r ∧ m) ∨ (s ∧ m). Since m is join irreducible, m = r ∧ m or m = s ∧ m, that is, m ≤ r or m ≤ s. For any r ∈ L, if there exists a s ∈ L such that r ∧ s = 0 and r ∨ s = 1, then r is called a complement of s. An order reversing involution : L → L is a mapping satisfying the conditions (r ) = r and r ≤ s =⇒ s ≤ r for all r, s ∈ L. If for all r ∈ L, r is a complement of r , then the order reversing involution : L → L is called a complementation mapping. The unit interval [0, 1] is a complemented and bounded complete lattice with the ordinary relation ≤. A complementation mapping : L → L has the following properties: Theorem  1.1.2Let {r j | j ∈ J } ⊆ L. Then r j . (i) ( r j ) = 

j∈J

(ii) (

j∈J



j∈J

r j ) =

j∈J

r j .

4

1 Introduction

Proof (i) Note that for all j ∈ J ,  j∈J

r j

≤(





r j ≤ r j and hence, we get r j ≤ (

j∈J

r j ) . Further, for all j ∈

j∈J

This implies that (



r j ) ≤

j∈J



J , r j

≤





r j ) and so



j∈J

r j

and then we have (

j∈J

r j . Finally, we obtain that (

j∈J

 j∈J

r j ) ≤

j∈J 

r j ) ≤ r j . r j .

j∈J

(ii) The proof is similar. Theorem 1.1.3 For a bounded and distributive lattice L, we have the following propositions: (i) If an element r ∈ L has a complement s, then s is unique. (ii) If every element of L has a complement, then the complementation mapping : L → L is also unique. Proof (i) Suppose that t is also a complement of r where t = s. Then s = 1 ∧ s = (r ∨ t) ∧ s = (r ∧ s) ∨ (t ∧ s) = 0 ∨ (t ∧ s) = (t ∧ s) and t = 1 ∧ t = (r ∨ s) ∧ t = (r ∧ t) ∨ (s ∧ t) = 0 ∨ (s ∧ t) = (s ∧ t). This yields that s = t. (ii) The proof is immediate by (i).

1.2 Completely Distributive Lattices Recall that a lattice L is called completely distributive if for all index set K , we have  k∈K j∈Jk

a kj =



aγk (k)

γ ∈C k∈K

 where C is the set of all choice functions γ : K → k∈K Jk such that γ (k) ∈ Jk ⊆ K and a kj ∈ L. Trivially, for a finite lattice, completely distributivity is equivalent to distributivity. If L is completely distributive, then every element of L can be written as a join of irreducible elements [5]. The following characterization of Raney [3] makes easy the proof of complete distributivity of a given lattice. Theorem 1.2.1 A complete lattice (L , ) is completely distributive if and only if for all r, s ∈ L, there exist x, y ∈ L such that (i) r  s =⇒ r  x and y  s, and (ii) ∀t ∈ L, either t  x or y  t. Using the theorem of Raney, we can easily prove the following theorem: Theorem 1.2.2 Let U be a non-empty domain of discourse. Then the family of all subsets of U , that is, the family P(U ) = {A | A ⊆ U } is completely distributive with the order of ordinary set inclusion.

1.2 Completely Distributive Lattices

5

Proof (P(U ), ⊆) has the following properties: If A, B ∈ P(U ) and A  B, then there exist C, D ∈ P(U ) such that A  C and D  B, and If E ∈ P(U ), E ⊆ C or D ⊆ E. Note that if A  B, then u ∈ A and u ∈ / B for some u ∈ U . If we get C = U \ {u} and D = {u}, we easily see that the first condition of Raney’s theorem holds. For the second condition, if E ⊆ U and E  U \ {u}, then we clearly have that D = {u} ⊆ E. As a result, the complete lattice (P(U ), ⊆) is completely distributive. L-Fuzzy Sets Let U be a non-empty domain of discourse and (L , ≤, ) be a complemented lattice. A mapping α : U → L is called a L-fuzzy subset of U [6]. We denote the family of all L-fuzzy subsets of U by F L (U ). The family F L (U ) is a complete lattice with the pointwise ordering ∀u ∈ U, α ≤ β ⇐⇒ α(u) ≤ β(u) for all α, β ∈ F L (U ). We denote the top and bottom elements of F L (U ) by 1 and 0, respectively where 1 : U → L , 1(u) = 1 for all u ∈ L, 0 : U → L , 0(u) = 0 for all u ∈ L where 1 and 0 are the top and bottom elements of L. The complement α of an L-fuzzy subset α is defined by α (u) = (α(u)) for all u ∈ U . Arbitrary joins and meets of a family of L-fuzzy subsets are defined as follows: ⊆ F L (U ), For any A  ( A)(u) = {α(u) | α ∈ A},   ( A)(u) = {α(u) | α ∈ A}   where and are the least upper bound and greatest lower bound for the subset {α(u) | α ∈ A} in L where u ∈ U . One of the important fuzzifications of a crisp point in fuzzy set theory is the fuzzy point defined by Wong in [7]. This concept provides many fundamental results due to local properties in fuzzy topological spaces [8]. As we will see in the sequel, in textures, we have special sets corresponding to fuzzy points and fuzzy co-points (see, Lemma 3.3.2). For λ ∈ L, with λ = 0, a fuzzy point u λ of F L (U ) is defined by u λ (z) =

λ, if z = u 0, if z = u

and for λ = 1, a fuzzy co-point u λ is defined by λ

u (z) =

λ, if z = u 1, if z = u

6

1 Introduction

for all z ∈ U , respectively. The notions of fuzzy point and fuzzy co-point are dual notions with respect to complementation:



(u λ ) (z) = (u λ (z)) =

λ , if z = u 1, if z = u



that is, (u λ ) = u λ and similarly, we have (u λ ) = u λ . Fuzzy lattices A lattice (L , ≤) is called a fuzzy lattice if it is a complete and completely distributive lattice with an order reversing involution . Example 1.2.3 (i) Powerset Fuzzy Lattice (P(U ), ⊆) The family P(U ) of all subsets of a domain of discourse U is a complete and completely distributive lattice (see, Theorem 1.2.2). (ii) Three valued fuzzy lattice L = {0, 21 , 1}. L is a bounded complete lattice with the order 0 < 21 < 1 where 0 and 1 are bottom and top elements of L, respectively. Clearly, it is completely distributive and has the order reversing involution defined by 1 = 0, 0 = 1 and ( 21 ) = 21 . This is equivalent to say that the function : L → L defined by r = 1 − r , where r = 0, 21 , 1 is a complementation mapping on L. (iii) L-Fuzzy lattice (F L (U ), ≤) Let us consider the family F L (U ) = {α | α : U → L is a mapping} of L-fuzzy sets. Since (L , ≤) is completely distributive, F L (U ) is also completely distributive. Indeed, if α, β ∈ F L (U ) and α  β, then there exists a u ∈ U such that α(u)  β(u) and α(u), β(u) ∈ L. Since L is completely distributive, by Raney’s theorem, there exist u 1 , u 2 ∈ L such that α(u)  u 1 and u 2  β(u), and further, for all t ∈ U , we have t ≤ u 1 or u 2 ≤ t. Let us consider the L-fuzzy sets γ : U → L , ∀u ∈ U, γ (u) = u 1 , η : U → L , ∀u ∈ U, η(u) = u 2 . Then we clearly have that α  γ and η  β. Further, for any ζ ∈ F L (U ), if ζ  γ , then there exists a u ∈ U such that ζ (u)  γ (u) = u 1 . Since, L is completely distributive, again by Raney’s theorem, for all u ∈ U , η(u) = u 2 ≤ ζ (u). This implies that η ≤ ζ . That is, F L (U ) is completely distributive. (iv) Unit fuzzy lattice ([0, 1], ≤) One of the primary examples to fuzzy lattices is the unit interval with the ordinary order ≤. Let u, v ∈ [0, 1] and u  v. Since v < u, we can choose x, y ∈ [0, 1] such that v < y < x < u. Then we clearly have that u  x and y  v.

1.2 Completely Distributive Lattices

7

Further, if z ∈ [0, 1] and z  x, then y ≤ z. Hence, x and y satisfy the conditions of Raney’s Theorem. Therefore, ([0, 1], ≤) is a completely distributive lattice. (v) Fuzzy lattice F (U ) of fuzzy subsets Since ([0, 1], ≤) is a fuzzy lattice, by (iii), F (U ) = F[0,1] (U ) = {α | α : U → [0, 1] is a mapping} is also fuzzy lattice. However, independently, using Raney’s theorem, we can easily show that F (U ) is completely distributive. Let α, β ∈ F (U ) and α  β. Then β(u) < α(u) for some u ∈ U . Let us choose a λ ∈ [0, 1] where β(u) < λ < α(u). Then we clearly get that α  u λ and u λ  β where u λ , u λ ∈ F (U ) are fuzzy point and fuzzy co-point, respectively. Further, let γ ∈ F (U ). If γ  u λ , then λ < γ (u) and so we get u λ ≤ γ . This means that F (U ) holds the conditions given in Raney’s theorem. The order reversing involution, that is, the complementation α = 1 − α of α is given by α (u) = 1 − α(u) for all u ∈ U . A fuzzy point (fuzzy co-point) is also a fuzzy set, and it satisfies similar properties as a crisp point. For instance, every fuzzy subset can be written as a join of fuzzy points or as a meet of fuzzy co-points: Lemma 1.2.4 For every fuzzy set α ∈ F (U ), we have α=

 λ 0 such that a < f (x0 ) −  < f (x0 ) ≤ b. By the assumption, for some δ > 0 x0 − δ < x < x0 =⇒ f (x0 ) −  < f (x). Since f is increasing, the implication x ∈ (x0 − δ, x0 ] =⇒ f (x) ∈ ( f (x0 ) − , f (x0 )]

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7 Fuzzy Rough Sets

holds and so we conclude that f ((x0 − δ, x0 ]) ⊆ ( f (x0 ) − , f (x0 )] ⊆ (a, b] = B  . If we say that B = (x0 − δ, x0 ], the inclusion f (B) ⊆ B  completes the proof. Theorem 7.2.3 Let ([0, 1], τl ) be the lower limit topological space and f : [0, 1] → [0, 1] be a decreasing function. Then the following propositions are equivalent: (i) f is τl -τl continuous at x0 ∈ [0, 1]. (ii) f is lower semicontinuous, that is, for all  > 0, there exists a δ > 0 such that for all x ∈ [0, 1], x < x0 + δ =⇒ f (x) < f (x0 ) + . Proof It is similar to the proof of Theorem 7.2.2.



A function F : [0, 1] × [0, 1] → [0, 1] is called (i) increasing if ∀x0 , y0 , x1 , y1 ∈ [0, 1], x0 ≤ x1 and, y0 ≤ y1 =⇒ F(x0 , y0 ) ≤ F(x1 , y1 ), (ii) increasing in the first component if for every fixed y0 ∈ [0, 1] and for all x, x  ∈ [0, 1], x ≤ x  =⇒ F(x, y0 ) ≤ F(x  , y0 ), (iii) decreasing in the first component if for every fixed y0 ∈ [0, 1] and for all x, x  ∈ [0, 1], x ≤ x  =⇒ F(x  , y0 ) ≤ F(x, y0 ). Definitions (ii) and (iii) given above can be similarly stated for the second component. Product Topology Let (U1 , τ1 ) and (U2 , τ2 ) be topological spaces. In a natural way, we may consider a topology on the Cartesian product U1 × U2 . Indeed, it is easy to see that the family {G 1 × G 2 | G 1 ∈ τ1 and G 2 ∈ τ2 } is a base for a topology denoted by τ1 × τ2 on U1 × U2 . The pair (U1 × U2 , τ1 × τ2 ) is called a product topological space and τ1 × τ2 is called product topology or box topology on U1 × U2 .

7.2 Continuity of Fuzzy Logical Connectives

149 p

Let us consider the product topology τu × τu = τu on [0, 1]2 where τu is the upper limit topology on [0, 1]. Theorem 7.2.4 Let F : [0, 1] × [0, 1] → [0, 1] be an increasing function and (x0 , y0 ) ∈ [0, 1] × [0, 1]. Then the following statements are equivalent: p

(i) F is τu -τu continuous at (x0 , y0 ). (ii) F is upper semicontinuous at (x0 , y0 ) ∈ [0, 1]2 , that is, for each  > 0 there exists a δ > 0 such that (x, y) ∈ (x0 − δ, x0 ] × (y0 − δ, y0 ] =⇒ F(x0 , y0 ) −  < F(x, y). Proof (i) =⇒ (ii): Let  > 0 be given and let F(x0 , y0 ) ∈ (a, b] ∈ B. Since  is a sufficiently small number, we have a < F(x0 , y0 ) −  < F(x0 , y0 ) ≤ b. Note that we also have F(x0 , y0 ) ∈ (F(x0 , y0 ) − , F(x0 , y0 )] ∈ B. Hence, by the assumption, there exists a (c, d]×]e, f ] ∈ Bup with (x0 , y0 ) ∈ (c, d]×]e, f ] such that F((c, d] × (e, f ]) ⊆ (F(x0 , y0 ) − , F(x0 , y0 )]. Since x0 ∈ (c, d], there exists a δ1 > 0 such that c < x0 − δ1 < x0 < d. Further, Since y0 ∈ (e, f ], there exists a δ2 > 0 such that e < y0 − δ2 < y0 < f . Let δ = min{δ1 , δ2 }. Therefore, we also have that F((x0 − δ, x0 ]×]y0 − δ, y0 ]) ⊆ (F(x0 , y0 )) − , F(x0 , y0 )]. This means that (x, y) ∈ (x0 − δ, x0 ] × (y0 − δ, y0 ], and this follows that F(x, y) ∈ (F(x0 , y0 ) − , F(x0 , y0 )]. As a result, we conclude the implication (x, y) ∈ (x0 − δ, x0 ] × (y0 − δ, y0 ] =⇒ F(x0 , y0 ) −  < F(x, y) as desired. (ii) =⇒ (i): Let B  = (a, b] ∈ Bu and F(x0 , y0 ) ∈ (a, b]. Then there exists an  > 0 such that a < F(x0 , y0 ) −  < F(x0 , y0 ) ≤ b. By the assumption, for some δ > 0 (x, y) ∈ (x0 − δ, x0 ] × (y0 − δ, y0 ] =⇒ F(x0 , y0 ) −  < F(x, y).

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7 Fuzzy Rough Sets

Since F is increasing, if (x, y) ∈ (x0 − δ, x0 ] × (y0 − δ, y0 ], then F(x, y) ∈ (F(x0 , y0 ) − , F(x0 , y0 )] and so F((x0 − δ, x0 ] × (y0 − δ, y0 ]) ⊆ (F(x0 , y0 ) − , F(x0 , y0 )] ⊆ (a, b] = B  . If we say that B = (x0 − δ, x0 ] × (y0 − δ, y0 ], then F(B) ⊆ B  completes the proof. Theorem 7.2.5 Let F : [0, 1] × [0, 1] → [0, 1] be a decreasing function and (x0 , y0 ) ∈ [0, 1] × [0, 1]. The following statements are equivalent: (i) F is τ pl -τl continuous at (x0 , y0 ). (ii) F is lower semicontinuous at (x0 , y0 ) ∈ [0, 1]2 , that is, for each  > 0 there exists a δ > 0 such that (x, y) ∈ [x0 , x0 + δ) × [y0 , y0 + δ) =⇒ F(x, y) < F(x0 , y0 ) + . Sequential Continuity and Left Continuity Note that in a topological space (U, τ ), a sequence (u n )n∈N converges to an element u ∈ U if for all G ∈ τ with u ∈ G, there exists a number N () ∈ N such that ∀n, n ≥ N () =⇒ xn ∈ G. Then we write lim {u n | n ∈ N} = u or, in brief, lim u n = u. For example, let n→∞

n→∞

(u n )n∈N be a sequence and suppose that it converges to a point u in the upper limit topological space ([0, 1], τu ). Then for all (a, b] ∈ τu with u ∈ (a, b], there exists a number N ∈ N such that the implication n ≥ N =⇒ u n ∈]a, b] holds. Note that for all sufficiently small number  > 0, we also have that u n ∈ (u − , b], that is, u −  < u n . For the lower limit topological space ([0, 1], τl ), if (u n )n∈N is a sequence and it converges to a point u in U , then we obtain that for all  > 0, u n < u + . Now, let (U1 , τ1 ) and (U2 , τ2 ) be topological spaces, f : U1 → U2 be a function and u ∈ U1 . Then f is said to be sequentially continuous at u ∈ U1 if for all sequence (u n )n∈N in U , the implication lim {u n | n ∈ N} = u =⇒ lim { f (u n ) | n ∈ N}) = f (u)

n→∞

n→∞

holds. Sequentially continuous functions preserve the convergence of sequences. In general, sequential continuity is weaker than continuity. Theorem 7.2.6 If f : U1 → U2 is τ1 -τ2 continuous, it is also sequentially continuous. Let (U, τ ) be a topological space and B be a base for τ . A subfamily Bu ⊆ B is called a local base at the point u if

7.2 Continuity of Fuzzy Logical Connectives

151

for all B ∈ B with u ∈ B, there exists a Bu ∈ Bu such that u ∈ Bu ⊆ B. Then (U, τ ) is called first countable if for all u ∈ U , there exists a countable local base. Note that if B is a base for τ , then the family Bu = {B ∈ B | u ∈ B} is a local base at u ∈ U . For instance, the topological space ([0, 1], τl ) is first countable. Indeed, for a point u ∈ U , the family Bu = {[u, u − 1/n[| n ∈ N} is a countable local base at u for the lower limit topology τl . Similarly, the upper limit topology τu is also first countable. Theorem 7.2.7 Let (U1 , τ1 ) and (U2 , τ2 ) be any two topological spaces and f : U1 → U2 be a function. If (U1 , τ1 ) is first countable, then the following statements are equivalent: (i) f : U1 → U2 is τ1 -τ2 continuous at u ∈ U1 . (ii) f is sequentially continuous at u ∈ U1 . Now, let F : [0, 1] × [0, 1] → [0, 1] be a mapping. For any y0 ∈ [0, 1], let us consider the function f y0 : [0, 1] → [0, 1] defined by ∀x ∈ [0, 1], f y0 (x) = F(x, y0 ). By the same way, for any x0 ∈ [0, 1], let us take the function f x0 : [0, 1] → [0, 1] defined by ∀y ∈ [0, 1], f x0 (x) = F(x0 , y). Since ([0, 1], τu ) and ([0, 1], τl ) are first countable, we may give the following two corollaries which are immediate results of Theorem 7.2.7. Corollary 7.2.8 Let f x0 and f y0 be increasing. Then (i) f y0 : ([0, 1], τu ) → ([0, 1], τu ) is sequentially continuous if and only if it is upper semicontinuous. (ii) f x0 : ([0, 1], τu ) → ([0, 1], τu ) is sequentially continuous if and only if it is upper semicontinuous. Corollary 7.2.9 Let f x0 and f y0 be decreasing. (i) The function f x0 : ([0, 1], τl ) → ([0, 1], τl ) is sequentially continuous if and only if it is lower semicontinuous. (ii) f y0 : ([0, 1], τl ) → ([0, 1], τl ) is sequentially continuous if and only if it is lower semicontinuous.

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7 Fuzzy Rough Sets

Left Continuity A function F : [0, 1] × [0, 1] → [0, 1] is called (i) left-continuous (right-continuous) [7] if lim {F(ηn , μn ) | n ∈ N} = F( lim {ηn | n ∈ N}, lim {μn | n ∈ N})

n→∞

n→∞

n→∞

for all non-decreasing (resp., non-increasing) sequences (ηn )n∈N , (μn )n∈N in [0, 1]. (ii) left-continuous (right-continuous) [7] in the first component if lim {F(ηn , μ) | n ∈ N} = F( lim {ηn | n ∈ N}, μ),

n→∞

n→∞

for all η, μ ∈ [0, 1] and non-decreasing (resp., non-increasing) sequences (ηn )n∈N , (μn )n∈N in [0, 1]. (iii) left-continuous (right-continuous) [7] in the second component if lim {F(η, μn ) | n ∈ N} = F(η, lim {μn | n ∈ N})

n→∞

n→∞

for all η, μ ∈ [0, 1] and non-decreasing (resp., non-increasing) sequences (ηn )n∈N , (μn )n∈N in [0, 1]. Note that if f y0 : ([0, 1], τu ) → ([0, 1], τu ) is sequentially continuous, then for every sequence (xn )n∈N ⊆ [0, 1], lim {F(xn , y0 ) | n ∈ N}) = lim f y0 ({xn | n ∈ N})

n→∞

n→∞

= lim { f y0 (xn ) | n ∈ N} n→∞

= f y0 ( lim {xn | n ∈ N}) n→∞

= F( lim {xn | n ∈ N}, y0 ). n→∞

Therefore, the sequential continuity of f y0 in fact is the left continuity of the mapping F : [0, 1] × [0, 1] → [0, 1] in the first component where the sequences are increasing. The increasing condition of f y0 guarantees the existence of the limit of a sequence. Note that if (xn )n∈N is an increasing sequence in ([0, 1], τu ), then it is convergent. Indeed, if we say xn = a ≤ 1, then there exists a number N such that for n ≥ N , xn ∈]c, a] ∈ τu for some c ∈ [0, 1]. That is, lim {xn | n ∈ N} = a. This n→∞ argument can be easily extended to the right continuity. The first countability is productive. Therefore, the product of any two first countable topological spaces is also first countable. This implies that the product space ([0, 1] × [0, 1], τu × τu ) is also first countable. Thus, by Theorems 7.2.4, and 7.2.5 we may give the following.

7.2 Continuity of Fuzzy Logical Connectives

153

Corollary 7.2.10 Let F : [0, 1] × [0, 1] → ([0, 1] be increasing mapping. p

(i) F : ([0, 1] × [0, 1], τu ) → ([0, 1], τu ) is left-continuous if and only if F is upper semicontinuous. p

(ii) F : ([0, 1] × [0, 1], τl ) → ([0, 1], τl ) is right-continuous if and only if F is lower semicontinuous. Theorem 7.2.11 Let F : [0, 1] × [0, 1] → [0, 1] be an increasing mapping. Then the following statements are equivalent. p

(i) F is left-continuous (τu -τu continuous) at (x0 , y0 ). (ii) F is left-continuous in each component. In other words, the functions f x0 , f y0 : [0, 1] → [0, 1] are left-continuous (τu -τu continuous). Proof Necessity: If F is left-continuous at (x0 , y0 ), taking the constant sequences (μn )n∈N = (y0 )n∈N and (ηn )n∈N = (x0 )n∈N , we clearly see that the functions f x0 , f y0 are left-continuous. Sufficiency. Let F be left-continuous in each component and  > 0 be given. Take a sequence (ηn , μn )n∈N converging (x0 , y0 ) in ([0, 1] × [0, 1], τ pu ). Note that the sequences (ηn )n∈N and (μn )n∈N also converge to x0 and y0 , respectively, in ([0, 1], τu ). Let us consider the increasing sequences (βn )n∈N and (γn )n∈N such that βn ≤ ηn and γn ≤ μn where lim βn = x0 and lim γn = y0 in ([0, 1], τu ). Since F is left-continuous in n→∞ n→∞ the second component, we have lim {F(x0 , γn ) | n ∈ N} = F(x0 , lim {γn | n ∈ N}) = F(x0 , y0 ).

n→∞

n→∞

Then there exists a number N such that for all n ≥ N , F(x0 , γn ) ∈]F(x0 , y0 ) − , 1] ∈ τu , that is, for all n ≥ N ,

F(x0 , y0 ) −  < F(x0 , γn ).

Further, since F is also left-continuous in the first component, for every fixed n ∈ N, lim {F(βm , γn ) | m ∈ N} = F( lim {βm | m ∈ N}, γn ) = F(x0 , γn ).

m→∞

m→∞

Then there exists a number M such that for all m ≥ M, F(βm , γn ) ∈ (F(x0 , γn ) − , 1] ∈ τu and hence,

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7 Fuzzy Rough Sets

F(x0 , γn ) −  < F(βm , γn ). The function F is increasing, and this implies that F(x0 , y0 ) − 2 < F(x0 , γn ) −  < F(βm , γn ) < F(ηm , μn ). Finally, for K = max{M, N } and for all k ≥ K , we conclude that F(x0 , y0 ) − 2 < F(ηk , μk ). That is, for the arbitrarily chosen , there exists a number K in N such that if k ≥ K , then F(ηk , μk ) ∈ (F(x0 , y0 ) − 2, 1]. As a result, F is left-continuous at (x0 , y0 ). The following results give the connection between continuity and distributivity [8]. Lemma 7.2.12 Let F : [0, 1] × [0, 1] → [0, 1] be a binary operation. (i) Let F be increasing in the first component. Then (a) F is left-continuous in the first component iff F is infinitely join-distributive in the first component, that is, for all η, μk ∈ [0, 1] and index set K = ∅,   {F(μk , η) | k ∈ K } = F( {μk | k ∈ K }, η). (b) F is right-continuous in the first component iff F is infinitely meet-distributive in the first component, that is, for all η, μk ∈ [0, 1] and index set K = ∅,

{μk | k ∈ k}, η).

{F(μk , η) | k ∈ K } = F(

(ii) Let F be decreasing in the first component. Then (a) F is left-continuous in the first component iff F is infinitely meet-distributive in the first component, that is, for all η, μk ∈ [0, 1] and index set K = ∅, {F(ηk , μ) | k ∈ K } = F( {ηk | k ∈ k}, μ). (b) F is right-continuous in the first component iff F is infinitely join-distributive in the first component, that is, for all η, μk ∈ [0, 1] and index set K = ∅,   {F(ηk , μ) | k ∈ K } = F( {ηk | k ∈ K }, μ). Further, we obtain similar equivalences if one replaces “in the first component” by “in the second component”.

7.2 Continuity of Fuzzy Logical Connectives

155

Proof We give only the proof of (i)(a). For the necessity, let F be left-continuous in [0, 1] and η ∈ [0, 1]. Since {μk | k ∈ the first component. Let us take {μk | k ∈ K } ⊆ K } is bounded, it has a least upper bound, say k∈K μk = μ. Suppose that for each n ∈ N, there exists a μk(n) ∈ {μk | k ∈ K } \ {μ} such that μ − μk(n) < 1/n. Note that for n + 1, μ − μk(n+1) < 1/(n + 1) and so μk(n) − μk(n+1) < −1/(n + 1) < 0. This means that for all n ∈ N, μk(n) ≤ μk(n+1) , that is, the sequence (μk(n) )n∈N is non-decreasing. Therefore, we get  μk(n) = μ lim μk(n) = n→∞

k∈K

and by the choice of the terms of the sequence (μk(n) )n∈N , for all n ∈ N, μk(n) < μ (note that (μk(n) )n∈N is convergent with respect to upper limit topology τu on [0, 1]). Since F is left-continuous in the first component, we also conclude that F(



μk , η) = F( lim μk(n) , η) = lim F(μk(n) , η) n→∞

k∈K

=



n→∞

F(μk(n) , η) =

n∈N



F(μk , η),

k∈K

as desired. Now, suppose that for some n ∈ N, ∀k ∈ K , μ − μk > 1/n.

This implies that for all k, μk < μk + 1/n < μ, and therefore, μk = μk0 for some k0 ∈ K . Thus, we conclude that   μk , η) = F(μk0 , η) ≤ F(μk , η). F( k∈K

k∈K

On since F is increasing, F(μk , η) ≤ F( the other hand, F(μ , η) ≤ F( k k∈K k∈K μk , η).

k∈K

μk , η) whence

For the sufficiency, let (μk )k∈N be an increasing sequence in [0, 1]. Then (μk )k∈N is convergent with respect to upper limit topology τu . If lim μk = μ, then k∈N μk = μ. This follows that

k→∞

 F( lim μk , η) = F(μ, η) = F( {μk | k ∈ K }, η) k→∞  = {F(μk , η) | k ∈ K } = lim {F(μk , η) | k ∈ N}. k→∞

156

7.3

7 Fuzzy Rough Sets

Textural Fuzzy Direlations

Through Brown’s Representation Theorem, we obtain a natural counterpart of a direlation between fuzzy textures (WU , WU ) and (WV , WV ) called textural fuzzy direlation which are in fact L-fuzzy subsets of the set U × M L × V where U and V are any two domains of discourse and M is a set of all molecules of L [9]. In this section, we take L = [0, 1] and we focus on the fuzzy lattice F (U ) whose elements are mappings α : U → [0, 1]. Fuzzy direlations can be also regarded as the mappings between fuzzy lattices F (U ) and F (V ). The arguments presented in Chap. 4 will be easily translated to the remarkable results for the fuzzy approximation spaces. A fuzzy relation ϕ is a mapping from U × V to [0, 1] and the family of all fuzzy relations is denoted by F (U × V ). A t-fuzzy direlation (φ, ) is a pair of mappings from U × (0, 1] × V to [0, 1] satisfying certain conditions. Essentially, this concept arosed from the textural arguments and is formally obtained via Brown’s Representation Theorem (see Theorem 3.1.4). The mappings φ and are the fuzzy subsets of U × (0, 1] × V , that is, φ, ∈ F (U × (0, 1] × V ). A t-fuzzy direlation can be regarded as a fuzzy version of direlations between textures. Besides that, it provides reasonable operators between the lattices of fuzzy subsets; t-fuzzy direlations establish a reasonable connection to fuzzy rough set theory. As in the study of algebraic properties of fuzzy logical connectives, the notion of continuity is an indispensable tool for a natural link between the fuzzy relations and t-fuzzy direlations. Proposition 7.3.1 Let (U, U), (V, V) and (W, W) be texture spaces. Then ((U × V ) × W, (U ⊗ V) ⊗ W) ∼ = (U × (V × W ), (U ⊗ (V ⊗ W)). Proof For the sake of shortness, we denote the product textures ((U × V ) × W, (U ⊗ V) ⊗ W) and (U × (V × W ), U ⊗ (V ⊗ W)) by (S, S) and (T, T ), respectively. Then the function ψ : S → T defined by ∀((u, v), w) ∈ S, ψ(((u, v), w)) = (u, (v, w)) ∈ T is one-to-one and onto. Further, the mapping ψ : S → T , A → ψ(A), A ∈ S is also one-to-one and onto. Hence, ψ is a textural isomorphism. Let us consider the fuzzy textures (WU , WU ) = (U × (0, 1], P(U ) ⊗ M) and

7.3 Textural Fuzzy Direlations

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(WV , WV ) = (V × (0, 1], P(V ) ⊗ M) where M = {(0, r ] | r ∈ [0, 1]} and M = (0, 1]. Let (r, R) be a direlation from (WU , WU ) to (WV , WV ). By definition of a direlation, we clearly have that r, R ∈ P(WU ) ⊗ WV = (P(U × (0, 1])) ⊗ (P(V ) ⊗ M). Note that by Proposition 7.3.1, P(U × (0, 1]) ⊗ (P(V ) ⊗ M) ∼ = (P(U × (0, 1]) ⊗ P(V )) ⊗ M = P(U × (0, 1] × V ) ⊗ M. This means that we have r, R ∈ P(U × (0, 1] × V ) ⊗ M up to isomorphism. In view of Brown’s Representation Theorem, F (U × (0, 1] × V ) is the fuzzy lattice corresponding to texture P(U × (0, 1] × V ) ⊗ M: F (U )

F (U × (0, 1] × V )

I

I

P(U ) ⊗ M

P(U × (0, 1] × V ) ⊗ M

Hence, for the direlation (r, R), there exist φr , R ∈ F (U × (0, 1] × V ) such that I(φr ) = r and I( R ) = R. Note that φr and R are the mappings from U × (0, 1] × V to [0, 1]. A fuzzy point (u, λ, v)μ and fuzzy co-point (u, λ, v)μ in F (U × (0, 1] × V ) may be given as  











(u, λ, v)μ (u , λ , v ) =

0, if (u, λ, v) = (u  , λ , v  ) 

and μ

(u, λ, v) (u , λ , v ) =

μ, if (u, λ, v) = (u  , λ , v  )

μ, if (u, λ, v) = (u  , λ , v  ) 1, if (u, λ, v) = (u  , λ , v  ),

respectively. The elements of the texture P(U × (0, 1] × V ) ⊗ M have the form I(φr ) = {(u, λ, v)μ | (u, λ, v)μ ≤ φr } = {((u, λ, v), μ) | μ ≤ φr (u, λ, v)} = {((u, λ), (v, μ)) | μ ≤ φr (u, λ, v)} for every φr ∈ F (U × (0, 1] × V ). Similarly, for the mapping R , we also have that I( R ) = {((u, λ), (v, μ)) | μ ≤ R (u, λ, v)}.

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Let us denote the p-sets and q-sets of the texture P(U × (0, 1] × V ) ⊗ M by P ((u,λ),(v,μ)) and Q ((u,λ),(v,μ)) . It is easy to see that P ((u,λ),(v,μ)) = {(u, λ)} × ({v} × (0, μ]) and



  Q ((u,λ),(v,μ)) = (WU \ {(u, λ)} × WV ∪ (WU × (0, μ] .

Lemma 7.3.2 P ((u,λ),(v,μ)) = I((u, λ, v)μ ) and Q ((u,λ),(v,μ)) = I((u, λ, v)μ ). Proof By Lemma 3.3.2, the proof is immediate.



Corollary 7.3.3 Let (r, R) be a direlation from (WU , WU ) to (WV , WV ). (i) r  Q ((u,λ),(v,μ)) if and only if μ < φr (u, λ, v). (ii) P ((u,λ),(v,μ))  R if and only if μ > R (u, λ, v). Proof (i) Let r  Q ((u,λ),(v,μ)) . Since I is a lattice isomorphism, by Lemma 7.3.2, we have I(φr )  I((u, λ, v)μ ) where I(φr ) = r and I((u, λ, v)μ ) = Q ((u,λ),(v,μ)) . Since I is a lattice isomorphism, we also have φr  (u, λ, v)μ , that is, (u, λ, v)μ < φr . Then we obtain that μ < φr (u, λ, v). For the sufficiency, if μ < φr (u, λ, v), then we easily find that r  Q ((u,λ),(v,μ)) . (ii) Let P ((u,λ),(v,μ))  R. Again by Lemma 7.3.2, I((u, λ, v)μ )  I( R ) where I( R ) = R and I((u, λ, v)μ ) = P ((u,λ),(v,μ)) . This implies that (u, λ, v)μ )  R and so R < (u, λ, v)μ . This gives that  R (u, λ, v) < μ. The sufficiency is similar. Now, we need the following characterizations. Theorem 7.3.4 Let r, R ∈ P(WU ) ⊗ WV . Then we have the following equivalences:  (i) r is a relation if and only if φr (u, λ, v) = {φr (u, λ , v) | 0 < λ < λ}. (ii) R is a corelation if and only if R (u, λ, v) = { R (u, λ , v) | λ < λ ≤ 1}.

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Proof (i) Let r be a relation. Suppose that  φr (u, λ, v) = {φr (u, λ , v) | 0 < λ < λ}. If φr (u, λ, v) < {φr (u, λ , v) | 0 < λ < λ}, then for some λ ∈ (0, 1] with 0 < λ < λ, we have φr (u, λ, v) < φ(u, λ , v). Let μ = φr (u, λ, v). Then by Corollary 7.3.3 (i), for μ < φr (u, λ , v), we get r  Q ((u,λ ),(v,μ)) . Further, λ < λ implies that Pλ = (0, λ]  Q λ = (0, λ ]. Hence, by Example 4.1.3, we have that P(u,λ)  Q (u,λ ) . Then by R1, we obtain r  Q ((u,λ),(v,μ)) . Again, by Corollary 7.3.3 (i), we find that μ < φr (u, λ, v) which is an immediate contradiction. Hence, the inequality  {φr (u, λ , v) | 0 < λ < λ} ≤ φ(u, λ, v) holds. Now, suppose that  {φr (u, λ , v) | 0 < λ < λ} < φr (u, λ, v). Let us choose a μ ∈ (0, 1] such that  {φr (u, λ , v) | 0 < λ < λ} < μ < φr (u, λ, v). By Corollary 7.3.3 (i), μ < φr (u, λ, v) implies that r  Q ((u,λ),(v,μ)) . Then by R2, there exists a (u  , λ ) ∈ U × (0, 1] such that P(u,λ)  Q (u  ,λ ) and r  Q ((u  ,λ ),(v,μ)) . By Example 4.1.2, 0 < λ < λ and u = u  . Then we also have r  Q ((u,λ ),(v,μ)) whence μ < φr (u, λ , v) by Lemma 7.3.3 (i). This gives the contradiction  μ< {φr (u, λ , v) | 0 < λ < λ}. For the sufficiency, let r ∈ P(WU ) ⊗ WV and  φ(u, λ, v) = {φ(u, λ , v) | 0 < λ < λ}. We show that r is a relation. Let r  Q ((u,λ),(v,μ)) . By Corollary 7.3.3 (i), μ < φr (u, λ, v). If P(u,λ )  Q (u,λ) , then 0 < λ < λ and so by the assumption, we conclude that φr (u, λ, v) ≤ φ(u, λ , v) and so by Corollary 7.3.3 (i), we obtain that r  Q ((u,λ ),(v,μ)) . That is r holds the condition R1. For the condition R2, let r  Q ((u,λ),(v,μ)) . By Corollary 7.3.3 (i), the equality  {φr (u, λ , v) | 0 < λ < λ} μ < φr (u, λ, v) = holds. Then there exists a λ ∈ (0, 1] with 0 < λ < λ such that μ < φr (u, λ , v). Clearly P(u,λ)  Q (u,λ ) . By Corollary 7.3.3 (i), we obtain that r  Q ((u,λ ),(v,μ)) as desired, that is, the condition R2 holds. The proof of the item (ii) is dual and it is left to the interested reader.



Theorem 7.3.5 (i) The mappings φr and R are increasing for λ for a fixed u and v. (ii) For all (u, v) ∈ U × V, R (u, 1, v) = 1.

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Proof By definitions of φr and R , the proof is straightforward. F (U )

(φr , R )

F (V )

I

P(U ) ⊗ M



I (r, R)

P(V ) ⊗ M

Remark 7.3.6 As is seen in the above diagram, the mappings φr and R are the fuzzy subsets of U × (0, 1] × V and they correspond to the direlation (r, R) from the fuzzy texture P(U ) ⊗ M to P(V ) ⊗ M under the lattice isomorphism I. Therefore, the pair (φr , R ) can also be regarded as a morphism from F (U ) to F (V ). More generally, Theorem 7.3.4 leads us to the following definition. Definition 7.3.7 (i) A mapping φ ∈ F (U × (0, 1] × U ) defined by ∀(u, λ, v) ∈ U × (0, 1] × V, φ(u, λ, v) = {φ(u, λ , v) | 0 < λ < λ} is called a textural fuzzy relation or in brief, t-fuzzy relation from F (U ) to F (V ). (i) A mapping ∈ F (U × (0, 1] × U ) defined by  ∀(u, λ, v) ∈ U × (0, 1] × V, (u, λ, v) = { (u, λ , v) | λ < λ ≤ 1} is called a textural fuzzy corelation or in brief, t-fuzzy corelation from F (U ) to F (V ). The pair (φ, ) is called a t-fuzzy direlation from F (U ) to F (V ). Theorem 7.3.8 If (i, I ) is the identity direlation on P(U ) ⊗ W, then the corresponding t-fuzzy direlation on F (U ) is the pair (φi , I ) where   λ, if u = v λ, if u = v I (u, λ, v) = φi (u, λ, v) = 0, if u = v, 1, if u = v for all u, v ∈ U and λ ∈ (0, 1] . Proof Note that by Corollary 7.3.3 (i), i  Q ((u,λ),(v,μ)) if and only if μ < φi (u, λ, v). That is, by Corollary 4.1.11 (1) and Theorem 3.2.11 (ii), the equivalence u = v and μ < λ if and only if μ < φi (u, λ, v) holds. Then u = v and φi (u, λ, v) = λ. If λ < φi (u, λ, v), then we obtain an immediate contradiction λ < λ. For the case φi (u, λ, v) < λ, we may choose a μ ∈ (0, 1] with φi (u, λ, v) < μ < λ. However, again by the above equivalence, we obtain

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μ < φi (u, λ, v) which is also an immediate contradiction. Now let u = v. Then again by the same equivalence, for all μ ∈ (0, 1], the inequality φi (u, λ, v) ≤ μ holds. This means that φi (u, λ, v) = 0. The proof of the second equality is similar.  Definition 7.3.9 The pair (φi , I ) given in Theorem 7.3.8 is called the identity t-fuzzy direlation on F (U ). Definition 7.3.10 Let (φ, ) be a t-fuzzy direlation on F (U ). If φi ≤ φ and ≤ I , then (φ, ) is called a reflexive t-fuzzy direlation. Theorem 7.3.11 Let r, R and (h, H ) be direlations from (WU , WU ) to (WV , WV ). Then we have the following inclusions: (i) r ⊆ h if and only if φr ≤ φh . (ii) R ⊆ H if and only if R ≤ H . Proof (i) Let h ⊆ r and φr  φh . Then there exists a (λ, v, μ) ∈ U × (0, 1] × V such that φh ((u, λ), v) < μ < φr (u, λ, v). Then by Corollary 7.3.3(i), we get r ⊆ Q ((u,λ),(v,μ)) and h  Q ((u,λ),(v,μ)) . However, this gives the contradiction r  h. The proof of the necessity and the item (ii) is similar. Theorem 7.3.12 Let (φ, ) be a t-fuzzy direlation on F (U ). Then (φ, ) is reflexive if and only if (r, R) is reflexive where (r, R) is the direlation on (WU , WU ) where φ = φr and = R . Proof The proof is immediate by Theorem 7.3.11. Now, let (r, R) be a direlation from (WU , WU ) to (WV , WV ). We know that by Lemma 4.1.16, the inverse (r, R)← = (R ← , r ← ) is also a direlation from (WV , WV ) to (WU , WU ). The following result gives a way to determine the inverse t-fuzzy direlation. Theorem 7.3.13 For the fuzzy relation (φ R ← , r ← ) from F (V ) to F (U ), ∀(v, μ, u) ∈ V × (0, 1] × U , the following equalities hold:  φ R ← (v, μ, u) = {λ ∈ (0, 1] | μ ≤ R (u, λ, v)}, and r ← (v, μ, u) = {λ ∈ (0, 1] | φr (u, λ, v) ≤ μ}.

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Proof Let us prove the first equality, leaving the second to the interested reader. By Lemma 4.1.23 (ii), P ((u,λ),(v,μ))  R if and only if R ←  Q (((v,μ),(u,λ)) . By Corollary 4.1.18, R ← is a relation and so by Corollary 7.3.3, μ > R (u, λ, v) if and only if λ < φ R ← (v, μ, u),

(1)

or equivalently, μ ≤ R (u, λ, v) if and only if φ R ← (v, μ, u) ≤ λ.

(2)

Since φ R ← is a fuzzy relation, we already know that φ R ← (v, μ, u) =



{φ R ← (v, μ , u) | 0 < μ < μ}.

By the equivalence (2), we clearly have that φ R ← (v, μ, u) ≤ {λ ∈ (0, 1] | μ ≤ R (u, λ, v)}. For the reverse inequality, suppose that there exists a λ such that φ R ← (v, μ, u) < λ < {λ ∈ (0, 1] | μ ≤ R (u, λ, v)}. Then by the right side of the above inequality, we have μ > R (u, λ , v) and so by  (1), we obtain that λ < φ R ← (v, μ, u) and this implies a contradiction. In view of Theorem 7.3.13, we obtain the following concept. Definition 7.3.14 Let (φ, ) be a fuzzy direlation from F (U ) to F (V ). Then the inverse of (φ, )← = ( ← , φ ← ) is defined by ← (v, μ, u) = {λ ∈ (0, 1] | μ ≤ (u, λ, v)},  {λ ∈ (0, 1] | φ(u, λ, v) ≤ μ}. φ ← (v, μ, u) = Remark 7.3.15 Note that by Theorem 7.3.13, the pair (φ, )← = ( ← , φ ← ) is a fuzzy direlation from F (V ) to F (U ).  Definition 7.3.16 Let (φ, ) be a t-fuzzy direlation on F (U ). If φ = ← and = φ ← , then (φ, ) is called a symmetric t-fuzzy direlation. Theorem 7.3.17 Let (φ, ) be a t-fuzzy direlation on F (U ). Then (φ, ) is symmetric if and only if (r, R) is symmetric where (r, R) is the direlation on (WU , WU ) where φ = φr and = R . Proof The proof is immediate by Theorem 7.3.11.

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Theorem 7.3.18 Let (r, R) and (d, D) be direlations from (WU , WU ) to (WV , WV ) and from (WV , WV ) to (W Z , W Z ), respectively. Then the pair (φd◦r , D◦R ) is a t-fuzzy direlation from F (U ) to F (Z ) where  {φd (v, μ, z) | v ∈ V, 0 < μ < φr (u, λ, v)} φd◦r (u, λ, z) = and D◦R (u, λ, z) =

{ D (v, μ, z) | v ∈ V, R (u, λ, v) < μ ≤ 1}.

Proof We prove the first equality. Let γ < φd◦r (u, λ, z). Then by Corollary 7.3.3 (i), d ◦ r  Q ((u,λ),(z,γ )) . By Definition 4.3.1, for all γ ∈ (0, 1], we clearly have the equivalence γ < φd◦r (u, λ, z) ⇐⇒ ∃(v, μ) ∈ WV with μ < φr (u, λ, v) and γ < φd (v, μ, z). Note that φd◦r is a t-fuzzy relation and so  φd◦r (u, λ, z) = {φd◦r (u, λ , z) | 0 < λ < λ}. Now, for 0 < λ < λ, let γ = φd◦r (u, λ , z). By the above equivalence, for γ , we get μ < φr (u, λ, v) and φd◦r (u, λ , z) < φd (v, μ, z). As a result, we obtain  φd◦r (u, λ, z) ≤ {φd (v, μ, z) | v ∈ V, 0 < μ < φr (u, λ, v)}. Suppose that φd◦r (u, λ, z)
μ. By the assumption, we have μ < β(v) and hence, β  v μ . This implies that I(β)  Q (v,μ) , and we obtain that I(β) ⊆ r → (I(α)). Now let us show that the equivalence given in (i) holds.

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Necessity: Let λ < α(u) =⇒ φr (u, λ, v) ≤ μ. Then by (2), we obtain β(v) ≤ μ. This gives that β(v) ≤ {μ ∈ (0, 1] | λ < α(u) =⇒ φr (u, λ, v) ≤ μ}. Now let β(v)