Mathematical Analysis of Some Rough Hybridized Models.Pdf

The University of New South Wales

School of Electrical Engineering and Computer Science and Engineering

Mathematical Analysis of Some Rough Hybridized Models

Tanzeela Shaheen Doctor of Philosophy 2016

Supervisor: Prof. Dr. Muhammad Shabir Assessor: Dr. Abc ii Contents

1 Preliminaries 1 1.1 Rough sets ...... 1 1.1.1 Basic concepts ...... 1 1.1.2 Rough membership functions ...... 7 1.1.3 Generalizations of rough sets ...... 9 1.2 Fuzzy sets ...... 12 1.2.1 Operations on fuzzy sets ...... 13 1.3 Hesitant fuzzy sets ...... 14

2 Fuzzi ed Rough Sets 18 2.1 Introduction ...... 18 2.2 Fuzzi ed rough approximations for indiscernible objects ...... 19 2.3 Proposed Algorithm ...... ...... 22 2.3.1 Illustrative Example ...... 23 2.4 Properties of fuzzi ed rough approximations ...... 25 2.5 Fuzzi ed rough membership function ...... 28 2.6 Measures associated with fuzzi ed rough sets ...... 30 2.7 Generalization of fuzzi ed rough sets using logical connectives . . . 34 2.7.1 Proposed Algorithm ...... 37 2.8 Comparative analysis ...... 39 2.9 Conclusion ...... 43

3 Generalized Hesitant Fuzzy Rough Sets 44 3.1 Introduction ...... 44 3.2 Approximations of Generalized hesitant fuzzy rough sets (GHF RS) . 44 3.2.1 (GHF RS) approximation operators and its properties . . . . . 45 3.2.2 Singleton GHF R approximation operators ...... 51 3.3 Topological structure associated with GHF R sets and singleton GHF R sets...... 57 3.4 Three way decision making with GHF RS model ...... 63 3.4.1 Proposed algorithm ...... 64 3.4.2 An illustrative example ...... 66 3.4.3 Comparative analysis ...... 69 3.5 Conclusion ...... 73

i CONTENTS ii

4 Multi-granulation structure of fuzzi ed rough sets and GHFRS 74 4.1 Introduction ...... ...... 74 4.2 Multi-granulation rough sets ...... 74 4.3 Multi-granulation fuzzi ed rough ( MGFR) sets ...... 76 4.3.1 Illustrative example ...... ...... 76 4.3.2 Properties of MGF RS ...... 77 4.4 Multi-granulation singleton generalized hesitant fuzzy rough sets . . . 82 4.5 Conclusion ...... 87

5 Graphical equivalence relations and graphical partitions 89 5.1 Introduction ...... 89 5.2 Graphical Equivalence Relation and Graphical Partition ...... 89 5.2.1 Graphs and Hypergraphs ...... 89 5.2.2 Relations on Hypergraphs ...... 92 5.2.3 Graphical equivalence relations(GERs) ...... 93 5.2.4 Graphical Partitions(GPs) ...... 96 5.2.5 Correspondence between graphical equivalence relations and graphical partitions ...... 99 5.3 Graphical Partitions as Surjective Mappings ...... 100 5.4 Two-tier Graphical Partitions ...... 102 5.5 Conclusion ...... 107

6 Rough graphs and their algebraic structure 108 6.1 Introduction ...... 108 6.2 Rough graphs ...... 108 6.3 Algebraic structure of rough graphs ...... 113 6.4 Conclusion ...... 120 Chapter 1

Preliminaries

1.1 Rough sets

Classical set theory has been extended to many valuable set theories due to the ongoing interests in knowledge representation and analysis of incompleteness and uncertainty in data analysis. Among other theories, rough set theory has proved to be a worthwhile extension. As a mathematical tool, theory of rough sets was initially proposed by Pawlak [46, 47] to handle imprecision and incompleteness in information systems. The initial approach adopted by Pawlak includes partitioning the universe set into granules (classes) of elements which are indistinguishable or indiscernible subject to the available information. Using these classes, an arbitrary subset of a universe is approximated by two de nable subsets called lower and upper approximations. The concealed knowledge in the information system can thus be revealed by using the concepts of upper and lower approximations in the rough set theory.

1.1.1 Basic concepts

In this section, the basic notions, de nitions, and properties of rough sets are recalled. An example has been presented to demonstrate these concepts. Let U be a nite set of objects and R U U be a given equivalence relation. The sets U; R are called the universe and an indiscernilbility relation, respectively. The indiscernibility relation represents our lack of knowledge about elements of U. A pair (U; R) is called an approximation space, where U is the universe and R is an equivalence relation on U. Let X be a subset of U, i.e. X U. Our goal is to characterize the set X with respect to R. In order to do this, we need additional notation and basic concepts of rough set theory which are presented below.

By [x]R we denote the equivalence class of R determined by element x. The indiscernilbility relation R describes - in a sense - our lack of knowledge about the universe

1 1. Preliminaries 2

U. Equivalence classes of the relation R, called granules, represent an elementary portion of knowledge we are able to perceive due to R. Using only the indiscernibility relation, in general, we are not able to observe individual objects from U but only the accessible granules of knowledge described by this relation. The concept of indiscernibility is central in rough set theory. Let I = (U; A) be an information system, where U is a non-empty nite set (the universe of discourse) and A is a non-empty nite set of attributes such that a : U Va for every a A. ! 2 Va is the set of values that attribute a may take. For any P A there is an associated equivalence relation R, also denoted by IND(P ), also called indiscernibility relation, given as:

R = IND(P ) = (x; y) U U : for all a P; a(x) = a(y) f 2 2 g If (x; y) IND(P ), then x and y are indiscernible by the attributes from P . The 2 partition of U, generated by R is denoted by U=R and the equivalence classes of the indiscernibility relation are denoted [x]R. Let X U. X can be approximated using only the information contained within R by constructing the lower and the upper approximations of X as below.

R(X) = x :[x]R X f g R(X) = x :[x]R X = f \ 6 ;g

Fig. 1.1. Rough set concept 1. Preliminaries 3

R is called the lower approximation operator and R is called the upper approximation operator. The positive, negative and boundary regions of a subset X of U are de ned by: POSR(X) = R(X)

NEGR(X) = U R(X) BNDR(X) = R(X) R(X) The positive region contains all objects of U that can be classi ed to the equivalence classes of U with respect to the equivalence relation R . The boundary region,

BNDR(X), is the set of objects that can possibly, but not certainly, be classi ed in this way. The negative region, NEGR(X), is the set of objects that cannot be classi ed to classes of U=R. Now we formally state the de nition of a rough set.

De nition 1.1.1 A set X is called rough (inexact) with respect to R if and only if the boundary region of X is non-empty, that is, BNDR(X) = : Otherwise it will be 6 ; a crisp (exact) set.

In case of decision systems, the set X to be approximated may be de ned by the given decision attribute(s). For example, if d is a given decision attribute having the domain yes, no then X may be de ned as: f g X = x U : d(x) = yes f 2 g Example 1.1.2 [61] Suppose we want to describe the classi cation of basic trac signs to a novice. We start by saying that there are three main classes of trac signs corresponding to

Warning (W), Interdiction (I), Order (O). Then, we say that these classes may be distinguished by such attributes as the Shape (S) and the principal color (PC) of the sign. Finally, we give a few examples of trac signs, like those shown in Table 1.1. These are

(a) sharp right turn,

1 (b) speed limit of 50 km h ; 1. Preliminaries 4

(d) go ahead.

Table 1.1. Examples of trac signs described by S and PC

Here "Class" is the decision attribute and S and PC are the condition attributes. Let R be the indiscernibility relation induced by S and PC. Then the sets of signs indiscernible by S and PC, that is, R, are as follows:

[a]R = a ; [b]R = b ; [c]R = [d]R = c; d : f g f g f g The lower and the upper approximations of the classes W= a , I= b,c and f g f g O= d are as below: f g R(W ) = a R(W ) = a f g f g R(I) = b R(I) = b; c; d f g f g R(O) = R(O) = c; d ; f g These approximations can be interpretted as below:

Class W includes sign (a) certainly and possibly no other sign, Class I includes sign (b) certainly and possibly signs (b), (c) and (d), Class O includes no sign certainly and possibly signs (c) and (d). 1. Preliminaries 5

The terms certainly and possibly refer to the absence or presence of ambiguity between the description of signs by S and PC from the one side, and by "Class", from the other side.

One can de ne the following four basic classes of rough sets, that is, four categories of vagueness:

1. A set X is roughly R de nable, if and only if R(X) = and R(X) = U. 6 ; 6 2. A set X is internally R unde nable, if and only if R(X) = and R(X) = U. ; 6 3. A set X is externally R unde nable, if and only if R(X) = and R(X) = U. 6 ; 4. A set X is totally R unde nable, if and only if R(X) = and R(X) = U. ; The intuitive meaning of this classi cation is the following. A set X is roughly R de nable means that with respect to R we are able to decide for some elements of U that they belong to X and for some elements of U that they belong to Xc. A set X is internally R unde nable means that with respect to R we are able to decide for some elements of U that they belong to Xc, but we are unable to decide for any element of U whether it belongs to X. A set X is externally R unde nable means that with respect to R we are able to decide for some elements of U that they belong to X, but we are unable to decide for any element of U whether it belongs to Xc. A set X is totally R unde nable means that with respect to R we are unable to decide for any element of U whether it belongs to X or Xc. A rough set X can be also characterized numerically by the following coecient.

De nition 1.1.3 Let (U; R) be an approximation space. Then R(X) R(X) = j j R(X) is called the accuracy of approximation , where X denotes the cardinality of j j X = : 6 ;

Obviously, 0 R(X) 1: If R(X) = 1 then X is crisp with respect to R (X is precise with respect to R), and otherwise, if R(X) < 1;X is rough with respect to R (X is vague with respect to R). It can be seen that the lower approximation R(X) is greatest de nable set contained in X, and R(X) is the least de nable set containing X with respect to the equivalence relation R: The other properties of lower and upper approximations operators have been listed in the following theorem. 1. Preliminaries 6

Theorem 1.1.4 Let (U; R) be an approximation space. For any X;Y U; the following properties hold for the lower and upper approximations.

1. R(X) X R(X); 2. X Y = R(X) R(Y ) and R(X) R(Y ); ) 3. R( ) = = R( ); ; ; ; 4. R(U) = U = R(U);

5. R(R(X)) = R(X) = R(R(X));

6. R(R(X)) = R(X) = R(R(X));

7. R(X Y ) = R(X) R(Y ); \ \ 8. R(X Y ) R(X) R(Y ); \ \ 9. R(X) R(Y ) R(X Y ); [ [ 10. R(X) R(Y ) = R(X Y ); [ [ 11. (R(X))c = R(Xc);

12. (R(X))c = R(Xc);

13. X is de nable R(X) = X R(X) = X R(X) = R(X); () () () 14. If X or Y are de nable, then R(X) R(Y ) = R(X Y ) and R(X Y ) = [ [ \ R(X) R(Y ): \ Properties 1, 2 and 5 of the above theorem show that R is an interior operator on U and is well-known that it induces a topology on U, that is, for any X U X if and only if R(X) = X 2 A topology de ned above is called the induced topology by R: The next two results have been taken from [29].

Proposition 1.1.5 Let R be an equivalence relation and be a topology induced by R. Then we have

R(X) = (R(Xc))c; that is; R is a closure operator on the topological space (U; ): 1. Preliminaries 7

The de nable sets can be characterized using topology as below.

Proposition 1.1.6 For every X U, the following conditions are equivalent. 1. X is de nable with respect to R;

2. X is an open subset in the topological space (U; ); where is the topology induced by R;

3. X is a closed subset in the topological space (U; ).

It follows from the above that the topological space (U; ) induced by an equivalence relation R on U has the property that for every subset X of U :

X is an open set if and only if it is a closed set.

1.1.2 Rough membership functions

In crisp set theory, characteristic function is employed to determine membership of an element to a given set subject to the available information. The interpretation of membership function in rough set theory is somehow di erent. It quanti es the degree of relative overlap between the set to be approximated and the equivalence class to which it belongs. In other words, it gives degree of belongingness of an element to a set subject to the given equivalence relation. Given an indiscernibilty relation R on U, the rough membership function R : U [0; 1] is de ned as follows X !

R [x]R X X (x) = j \ j [x]R j j where : denotes the cardinality. j j The rough membership function expresses conditional probability that x belongs to X given R and can be interpreted as a degree that x belongs to X in view of information about x expressed by R. The meaning of rough membership function can be depicted as shown in Fig. 1.2. 1. Preliminaries 8

[x]R

R X (x) = 0

[x]R

R 0 < X (x) < 1

[x]R

R X (x) = 1 Fig. 1.2. Membership of an element through rough membership function

The rough membership function can be used to de ne approximations and the boundary region of a set, as shown below:

R R(X) = x U : X (x) = 1 f 2 R g R(X) = x U : X (x) > 0 f 2 gR BNDR(X) = x U : 0 < (x) < 1 f 2 X g Theorem 1.1.7 For a given approximation space (U; R); the rough membership function has the following properties:

1. R (x) = 1 if and only if x R(X) X 2 2. R (x) = 0 if and only if x U R(X) X 2 R 3. o < (x) < 1 if and only if x BNDR(X) X 2 R R 4. c (x) = 1 (x) for any x U X X 2 R R R 5. X Y (x) max(X (x); Y (x)) for any x U [ 2 R R R 6. X Y (x) min(X (x); Y (x)) for any x U: \ 2 1. Preliminaries 9

1.1.3 Generalizations of rough sets

The rough set theory (RST ) proved to be a useful tool in supporting data-related tasks such as classi cation, decision making and description. Many extensions of rough set theory have been proposed by generalizing the equivalence relation, using multiple relations and using the subset operator. The illustration in g. 1.3 shows the main RST extensions in relation to the aspects of the theory they extend. Some of them are discussed brie y in below.

Rough set theory

subset operator• mapping based single relation based multiple relation based extensions extensions extensions based extensions

Variable•precision T•rough sets Generalized Multi•granulation rough sets rough sets rough sets

Dominance based rough sets

Characteristic relation based rough sets

Nbhd rough sets

Fig. 1.3. A taxonomy of rough set extensions

Generalized rough sets [83]

Given a binary relation R and two elements x; y U; if xRy; we say that y is R related 2 to x: A binary relation may be more conveniently represented by a mapping r : U ! 2U : r(x) = y U : xRy f 2 g 1. Preliminaries 10

That is, r(x) consists of all R-related elements of x: Following Pawlak's approach, we de ne two unary set-theoretic operators apr and apr :

apr(A) = x : r(x) A f g = x U :for all y U; xRy implies y A f 2 2 2 g apr(A) = x : r(x) A = f \ 6 ;g = x U : there exists a y U such that xRy and y A f 2 2 2 g The set apr(A) consists of those elements whose R-related elements are all in A; and apr(A) consists of those elements such that at least one of whose R related elements is in A. The pair (apr(A); apr(A)) is referred to as the generalized rough set of A induced by R: Operators apr; apr : 2U 2U are referred to as the generalized rough ! set operators. The set r(x) may be interpreted as a neighnorhood of x. Hence, apr and apr are indeed the interior and closure of A: In case where R is an equivalence relation, generalized rough set operators reduce to the operators in Pawlak rough set model. For an arbitray relation, generalized rough set operators do not necessarily satisfy all the properties in Pawlak rough set models. Nevertheless, following properties hold in any rough set model, that is, independent of the properties of the binary relation.

apr(U) = U; apr( ) = ; ; ; A B = apr(A) apr(B); apr(A) apr(B); ) apr(A) = (apr(Ac))c; apr(A) = (apr(Ac))c; apr(A B) = apr(A) apr(B); \ \ apr(A B) apr(A) apr(B); [ [ apr(A B) = apr(A) apr(B); [ [ apr(A B) apr(A) apr(B): \ \ In order to construct a rough set model so that the above properties hold, it is necessary to impose certain conditions on the binary relation R. In fact, each of these properties corresponds to a property of the binary relation. A relation R is a serial relation if for all x U there exists y U such that 2 2 xRy: A relation is a re exive relation if for all x U the relationship xRx holds. A 2 1. Preliminaries 11 relation is symmetric relation if for all x; y U; xRy implies yRx holds. A relation 2 is transitive relation if for three elements x; y; z U; xRy and yRz imply xRz: A 2 relation is Euclidean when for all x; y; z U; xRy and xRz imply yRz: Since the 2 approximation operators are de ned through the mapping r; it is more convenient to express equivalently the conditions on a binary relation as follows: serial: for all x U; r(x) = : 2 6 ; re exive: for all x U; x r(x): 2 2 symmetric: for all x; y U; if x r(y); then y r(x): 2 2 2 transitive: for all x; y U; if y r(x); then r(y) r(x): 2 2 Euclidean: for all x; y U; if y r(x); then r(x) r(y): 2 2 The rough set models are named according to the properties of the binary relation. For example, a rough set model constructed from a symmetric relation is referred to as a symmetric rough set model. The ve properties of a binary relation, namely, the serial, re exive, symmetric, transitive, and Euclidean properties, include ve properties of the approximation operator, stated as below.

relation type property holds serial: apr(A) apr(A) re exive: apr(A) A symmetric: apr(A) apr(apr(A)) transitive: apr(A) apr(apr(A)) Euclidean: apr(A) apr(apr(A)) By combining these properties, one can construct more rough set models. For instance, if R is re exive and symmetric, that is, R is a compatibility (also called tolerance) relation, we obtain the rough set model built using a compatibility relation. A compatibilty relation is a serial relation but not necessarily a transitive or an Euclidean relation.

T-rough sets [80]

Due to imprecise human knowledge, some times it is not possible to nd a suitable equivalence relation among the elements of the universe set U: Therefore, researchers felt the need to generalize the rough set theory. One of the generalizations was presented by Yamak et al. 1. Preliminaries 12

De nition 1.1.8 Let X and Y be two non-empty universes. Let T be a set valued mapping given by T : X P (Y ): Then the triplet (X;Y;T ) is referred to as a ! generalized approximation space. Any set valued function from X to P (Y ) de nes a binary relation by setting T = (x; y): y T (x) : Obviously, if is an arbitrary f 2 g binary relation from X to Y, then it can de ne a set valued mapping T : X P (Y ) ! by T(x) = y Y :(x; y) where x X: For any set A Y a pair of lower and f 2 2 g 2 upper approximations T (A) and T (A) are de ned by

T (A) = x X : T (x) A f 2 g and T (A) = x X : T (x) A = : f 2 \ 6 ;g The pair (T (A); T (A)) is referred to as a generalized rough set, and T and T are referred to as lower and upper generalized approximation operators, respectively.

Multi-granulation rough sets

The Pawlak's rough set theory is based on a single granulation. Using more than one binary relations, Qian et al. [50] presented the concept of multi-granulation rough sets which is stated below.

De nition 1.1.9 Let R1; R2; :::; Rn be n independent equivalence relations over a universe set X and A X: The lower and upper multi-granulation rough approximations of A in X are de ned respectively as

( Ri)(A) = x X [x]Ri A for some i = 1; 2; :::; n i=1 f 2 j g P and n

( Ri)(A) = x X [x]Ri A = for all i = 1; 2; :::; n : i=1 f 2 j \ 6 ; g The boundaryP region of A X under MGRS environment is de ned as n n Bnd n (A) = ( Ri)(A) ( Ri)(A): Ri i=1 n i=1 i=1 P P P 1.2 Fuzzy sets

Traditional set theory is based on binary, or two-valued, logic. Given a universe set U, a subset A of U can be de ned in several ways: Suppose that U is the set of integers. The subset of prime numbers less than 10 can be speci ed by listing its members: A = 2; 3; 5; 7 or by providing de ning properties f g A = x : x is a prime number less than 10 : f g 1. Preliminaries 13

Alternately, we de ne a subset A by its characteristic function, which is also denoted by the set name, A : U 0; 1 from U into the binary set 0; 1 given ! f g f g by 1 if x A A(x) = 2 ( 0 if x = A 2 Zadeh [86] de ned a fuzzy subset A of U as a function A : U [0; 1], that is, a ! characteristic function from U into the interval [0; 1]. The function from U to [0; 1] is called membership function of A, usually denoted by A: The value A(x) (or A(x)) is called the grade of membership of the point x in the fuzzy set A or the degree to which the point x belongs to the set A. For example, the fuzzy subset A of \the numbers close to 10" could be de ned by the membership function 1 (x) = A 1 + 5(x 10)2 Fig. 1.4. highlights the di erence between the concepts of membership in the crisp and fuzzy set theories.

a b

1 1

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0 0 0.5 1 0.5 1 universe of discourse universe of discourse Fig. 1.4. Membership function according to (a) crisp set theory (b) fuzzy set theory

1.2.1 Operations on fuzzy sets

The set of all fuzzy subsets of U will be denoted by (U). FP De nition 1.2.1 Let f; g (U). If f(x) g(x) for all x U, then f is said to 2 FP 2 be contained in g, and we write f g (or g f). 1. Preliminaries 14

Clearly, the inclusion relation is a partial order on (U). FP De nition 1.2.2 Let f; g (U): The union and intersection of fuzzy subsets f 2 FP and g of U for any x U is de ned respectively as follows: 2 (f g)(x) = f(x) g(x) [ _ (f g)(x) = f(x) g(x) \ ^ De nition 1.2.3 The complement of a fuzzy subset f (U) is denoted and de ned 2 FP as follows:

f 0(x) = 1 f(x); for all x U: 2 De nition 1.2.4 The fuzzy subsets of U, denoted by 0 and 1; de ned respectively as

0(x) = 0 1(x) = 1

for all x U; are called the empty (or null) fuzzy subset and the whole (or full) 2 fuzzy subset of U respectively.

1.3 Hesitant fuzzy sets

In this section we recall some basic notions related to the hesitant fuzzy sets. Xu et al. [78] de ned hesitant fuzzy sets in the following way:

De nition 1.3.1 Let U be a xed set. A hesitant fuzzy set (HFS) on U is given in terms of a function that when applied to U returns a subset of [0; 1] which can be represented as the following:

= (x; h (x)) : x U F f F 2 g where h (x) is a set of values in [0; 1], denoting the possible membership degrees F of the element x U to the set . For convenience, h (x) is called a hesitant fuzzy 2 F F element.

If h is a hesitant fuzzy set on U; then h(x) is a set of values in [0; 1] which represents the possible membership degrees of the element x U to the hesitant fuzzy set h: h(x) 2 will be called a hesitant fuzzy element (HFE) and its cardinality is called length of the HFE. Throughout, HF (U) will represent the collection of all hesitant fuzzy sets on U: 1. Preliminaries 15

Throughout, h(x) will be a nite subset for every hesitant fuzzy set h and any x U: 2 For a hesitant fuzzy element h(x); Torra [69] de ned its lower and upper bounds as below: lower bound: h (x) = min : h(x) ; f 2 g upper bound: h+(x) = max : h(x) : f 2 g Some related notions and operations on hesitant fuzzy elements by Yang et al. [81] are listed below.

De nition 1.3.2 Let x; y U and h1(x) and h2(y) be two hesitant fuzzy elements 2 on U: Then, their union ( ), intersection ( ) and containment ( ) are de ned in the Y Z following way:

1. Union: h1(x) Y h2(y) = h1(x) h2(y): max h1(x); h2(y) , f 2 [ f + g+ 2. Intersection: h1(x) Z h2(y) = h1(x) h2(y): min h1 (x); h2 (y) ; f 2 [ + f + g 3. Containment: h1(x) h2(y) h(x) h(y) and h (x) h (y): () 1 2 1 2 The above operations are de ned for hesitant fuzzy elements. For hesitant fuzzy sets, the following operations are de ned.

De nition 1.3.3 For hesitant fuzzy sets h, h1 and h2 on U; the following operations have been de ned:

1. Containment (Yang et al. [81]): h1 is contained in h2; denoted by h1 h2; if and only if h1(x) h2(x) for all x U; 2 2. Union (Torra [69]): union of h1 and h2 is the hesitant fuzzy set h1 dh2 de ned as (h1 h2)(x) = h1(x) h2(x) = h1(x) h2(x): max h(x); h(x) for all d Y f 2 [ f 1 2 g x U; 2 3. Intersection (Torra [69]): intersection of h1 and h2 is the hesitant fuzzy set + h1 h2 de ned as (h1 h2)(x) = h1(x) h2(x) = h1(x) h2(x): min h (x); e e Z f 2 [ f 1 h+(x) for all x U; 2 g 2 4. Complement (Torra [69]): complement of h is the hesitant fuzzy set hc which is de ned as (hc)(x) = 1 for all x U; [h(x)f g 2 2 5. Di erence (Liao [35]): di erence of h1 and h2 is the hesitant fuzzy set h1 h2 de ned as (h1 h2)(x) = ; where 1 h1(x); 2 h2(x)f g 2 S 2 1 2 if and = 1 1 2 1 2 2 = 6 ( 0 otherwise

Remark 1.3.4 1. The operations e; d are de ned on hesitant fuzzy sets while operations Z; Y are de ned on the corresponding hesitant fuzzy elements. We shall use the symbol c both to denote complements of hesitant fuzzy sets and hesitant fuzzy elements. 1. Preliminaries 16

2. Yang et al. in [81] proved that for hesitant fuzzy sets h1; h2 HF (U); h1 h2 2 e h1; h2 and h1; h2 h1 h2. d 3. Gang et al. proved in [20] that hesitant fuzzy union and intersection can also be expressed in the following form:

(h1 d h2)(x) = max 1; 2 ; 1 h1(x)[; 2 h2(x) f g 2 2 (h1 e h2)(x) = min 1; 2 : 1 h1(x)[; 2 h2(x) f g 2 2

4. By induction, for an arbitrary nite collection hi (i I = 1; 2; :::; n ) of hesitant 2 f g fuzzy sets, their union and intersection will be taken as follows:

( hi) = (((h1 h2) h3)::: hn) idI d d d 2 ( hi)(x) = (((h1 h2) h3)::: hn) ieI e e e 2 5. Equality of hesitant fuzzy sets will be taken as follows:

Two hesitant fuzzy sets h1 and h2 are equal if h1(x) = h2(x) for all x U: 2 It should be noted that the equality h1(x) = h2(y) for any x; y U is not followed by 2 h1(x) h2(y) and h2(y) h1(x): For example, if h1(x) = 0:3; 0:4; 0:7 and h2(y) = f g 0:3; 0:5; 0:6; 0:7 then h1(x) h2(y) and h2(y) h1(x): Nevertheless, h1(x) = h2(y): f g 6 Yang et al. [81] investigated the following operational laws for hesitant fuzzy elements.

Theorem 1.3.5 For any hesitant fuzzy elements h1(x); h2(y) and h3(z) on U, the following properties hold: 1. Idempotent: h1(x) Z h1(x) = h1(x); h1(x) Y h1(x) = h1(x); 2. Commutativity: h1(x) Z h2(y) = h2(y) Z h1(x); h1(x) Y h2(y) = h2(y) Y h1(x); 3. Associativity: h1(x) Z (h2(y) Z h3(z)) = (h1(x) Z h2(y)) Z h3(z); h1(x) Y (h2(y) Y h3(z)) = (h1(x) Y h2(y)) Y h3(z); 4. Distributivity: h1(x) Z (h2(y) Y h3(z)) = (h1(x) Z h2(y)) Y (h1(x) Z h3(z)); h1(x) Y (h2(y) Z h3(z)) = (h1(x) Y h2(y)) Z (h1(x) Y h3(z)); c c c c c 5. De Morgan's laws: (h1(x) Z h2(y)) = h1(x) Y h2(y); (h1(x) Y h2(y)) = h1(x) Z c h2(y); c c 6. Double negation: (h1(x)) = h1(x):

Torra [69] and Yang et al. [81] mentioned some special hesitant fuzzy sets on U.

1. empty set: h(x) = 0 for all x U; f g 2 2. full set: h(x) = 1 for all x in U; f g 1. Preliminaries 17

3. complete ignorance: for any x U; h(x) = [0; 1]; 2 4. set for a nonsense x : h(x) = for some x U; ; 2 5. constant set : h(x) = A for all x U; where A is some xed subset of [0; 1]: 2

Throughout, the empty hesitant fuzzy set will be denoted by h0 while full hesitant fuzzy set will be denoted by hU : To compare hesitant fuzzy elements, Xia and Xu [77] introduced score function. They made the following assumptions for this function: -The values of all the HFEs are arranged in an increasing order. -The HFEs have the same length when they are compared. Therefore, if any two HFEs have di erent lengths, the shorter one will be extended by adding the maximum element until both HFEs have the same length.

De nition 1.3.6 (Xia and Xu [77])For a hesitant fuzzy element h(x), score function is de ned as

s(h(x)) = #h(x) 0 1 h(x) 2X where s( ) is the score function and #@h(x) is theA number of elements in h(x):

Example 1.3.7 Let h1 = 0:3; 0:5; 0:8 and h2 = 0:1; 0:4; 0:9 be two hesitant fuzzy f g f g elements. Their union, intersection, complements and di erence have been calculated in below: h1 h2 = 0:3; 0:5; 0:8 0:1; 0:4; 0:9 Y f g Y f g = 0:3; 0:4; 0:5; 0:8; 0:9 f g h1 h2 = 0:3; 0:5; 0:8 0:1; 0:4; 0:9 Z f g Z f g = 0:1; 0:3; 0:4; 0:5; 0:8 f g hc = 0:2; 0:5; 0:7 ; hc = 0:1; 0:6; 0:9 1 f g 2 f g h1 h2 = 0:22; 0:44; 0:17; 0:78; 0:67; 0:0 f g Also, the scores of h1 and h2 are caluclated in below:

0:3+0:5+0:8 s(h1) = 3 = 0:53 0:1+0:4+0:9 s(h2) = 3 = 0:47

Here score of h1 is greater than the score of h2. Chapter 2

Fuzzi ed Rough Sets

2.1 Introduction

There may be some data set in which values of attributes are both symbolic and numerical. In these cases, it will be dicult to nd the underlying patterns from the classical rough set approach. If the attribute values in the given information system are not numerical, that is, they are qualitative rather than quantitative then it may be necessary to fuzzify the given universe set or the subset to be approximated. Also, in some situations clustering of elements subject to the given attributes may not be useful or even may not be possible. A slight di erence in the attribute value may lead two elements in di erent classes which, in real, may be very close to each other. Then, rather than considering elements' indistinguishability, assessing their similarity (to a certain degree) can be more suitable. Taking into account the diculty in handling the above situations, many authors introduced hybridization of fuzzy and rough set theories in di erent ways [39, 40, 41, 53]. From application point of view, constructive approach proves to be more useful which was initiated by Dubois and Prade [17, 18]. In this approach, lower and upper approximations are constructed by means of fuzzy relations. In the above stated paper, t-norm min and its dual conorm max were used to de ne fuzzy similarity relation. This approach was further generalized by Radzikowska and Kerre [52], by using an implication operator and a t-norm. They de ned a broad family of fuzzy rough sets with respect to a fuzzy similarity relation which they called ( ; ) fuzzy rough sets. This approach was further generalized I T by Ouyang et al. [43] in which they used transitivity instead of classical sup-min T transitivity of the fuzzy relation. These approaches of de ning fuzzy rough set use similarity (or similarity) relations in which the classes are disjoint. Less e ort T has been made to explore the structure of these fuzzy rough sets using intransitive fuzzy relations. Also, interpretation of the results depend highly on the particular

18 2. Fuzzi ed Rough Sets 19 fuzzy relation being used but the choice of appropriate fuzzy relation has not been addressed in them. This issue is addressed and resolved in this chapter. Some other generalizations can be seen in [3, 8, 10, 12, 43, 51, 54, 67, 68, 72, 76, 84]. In the present chapter, a new technique for fuzzi cation of rough set is introduced which involves both the fuzzi cation of the information system if the attribute values are linguistic terms and a tolerance (intransitive) fuzzy relation which is used to measure the compatibility in indiscernible objects; the objects which don't have exactly the same attributes but they are similar or compatible up to a certain degree . Upper and lower approximations are then de ned using the above concepts for approximating subsets of the given universe set. The technique is further extended by using implications and t-norms. For application purposes, two algorithms have been introduced for approximating subsets of the universe based on their similarity among objects, subject to the given attributes.

2.2 Fuzzi ed rough approximations for indiscernible objects

Many attempts have been made to fuzzify rough approximations. Two approaches have mainly been adopted; rst, to fuzzify the set to be approximated so that the objects meet its characteristics to varying degrees and second, to highlight similarity between objects through a fuzzy relation and categorizing the objects into granules with soft boundaries based on their similarity to one another. In this section, we have developed an algorithm for approximating subsets of the universe by combining both of these extended approaches. The algorithm is exible enough to handle both the numerical as well as linguistic attribute values in the information table. In some situations, there are some positive as well as negative attributes. Then the same attribute value may have di erent meaning for the two attributes. For example, the value "high" has exactly the opposite meaning regarding the attributes "hardworking" and "carelessness". Thus, to fuzzify the attribute values; the algorithm gives a margin to use separate fuzzy membership functions for positive and negative attributes. Also, since we are considering the objects which are all discernible, that is, no two objects have exactly the same attributes, categorizing the objects into classes with sharp boundaries will be of no use. Therefore, a fuzzy relation has been de ned to emphasize the relationship between the objects.

De nition 2.2.1 A fuzzy subset f (U U) is called a fuzzy binary relation or 2 F simply a fuzzy relation on U: A fuzzy relation f is called a serial fuzzy relation if for each x U; there exists y U such that f(x; y) = 1; f is referred to as a re exive 2 2 2. Fuzzi ed Rough Sets 20 fuzzy relation if f(x; x) = 1 for all x U; f is referred to as a symmetric fuzzy 2 relation if f(x; y) = f(y; x) for all x; y U; f is referred to as a transitive fuzzy 2 relation if f(x; z) (f(x; y) f(y; z)) for all x; y; z U. y _U ^ 2 A re exive, symmetric2 fuzzy relation on U is called proximity relation or compatibility relation in U (also called tolerance relation): These proximity relations can intuitively be interpreted as measures of `likeness' or `sameness' among the elements. When R is a fuzzy compatibility relation, compatibility classes are de ned in terms of a speci ed membership degree : An compatibility class is a subset A of U such that R(x; y) for all x; y A: 2 A fuzzy binary relation is called a similarity relation (or fuzzy equivalence relation) if it is a re exive, symmetric and transitive fuzzy relation. The notion of similarity relation is a natural generalization of an equivalence relation.

De nition 2.2.2 Let (U; R) be a fuzzy approximation space, where U is a nite (non- empty) set of objects and R is a fuzzy tolerance relation characterized by its membership function R : U U [0; 1]: For any (0; 1]; the fuzzi ed lower and upper rough ! 2 approximations for a given set X U are de ned as c R (X) = x U : R(x; y) < for all y X f 2 2 g and

R (X) = x U : R(x; y) for some y X : f 2 2 g The pair (R (X); R (X)) is referred to as fuzzi ed rough set. Positive, negative and boundary regions of X U for any (0; 1] are denoted and de ned as: 2

POSR(X) = R (X); c NEGR(X) = (R (X)) ;

BNDR(X) = R (X) R (X): Based on the given information table, the elements in the positive, negative and boundary region can be interpreted, respectively as certain members, certain non- members and possible members (may or may not be) of X up to a relational degree of (0; 1]: Throughout, R will be considered as tolerance fuzzy relation until otherwise 2 speci ed.

Example 2.2.3 Let U = x1; x2; x3; x4; x5; x6 be a universe set with fuzzy tolerance f g relation R as de ned in Table 2.1.

The lower and upper fuzzi ed approximations of a subset X = x1; x3; x5 with f g = 0:9 are calculated as below: 2. Fuzzi ed Rough Sets 21

R x1 x2 x3 x4 x5 x6 x1 1 0.5 0.9 0.6 0.2 0.3 x2 0.5 1 0.7 0.1 0.4 0.5 x3 0.9 0.7 1 0.3 0.7 0.8 x4 0.6 0.1 0.3 1 0.9 0.6 x5 0.2 0.4 0.7 0.9 1 0.8 x6 0.3 0.5 0.8 0.6 0.8 1

Table 2.1: A fuzzy tolerance relation

c Here X = x2; x4; x6 . Thus, R (X) = x1; x3 as R(x1; y) < 0:9 for all f g 0:9 f g y x2; x4; x6 and R(x3; y) < 0:9 for all y x2; x4; x6 : 2 f g 2 f g Also, as R(x1; x1) = 1 > 0:9;

R(x3; x3) = 1 > 0:9;

R(x4; x5) = 0:9;

R(x5; x5) = 1 > 0:9;

so, R0:9(X) = x1; x3; x4; x5 . Thus, ( x1; x3 ; x1; x3; x4; x5 ) is 0:9 fuzzi ed f g f g f g rough set. Also, POSR(X) = x1; x3 ;NEGR(X) = x2; x6 and BNDR(X) = f g f g x4; x5 : f g Remark 2.2.4 In the above de nition, signi es the level or degree up to which the relationship or compatibility among the objects is to be considered so that we may interpret lower approximation as a subset of U containing only those elements which are not related (up to to ) to any element outside X, that is, Xc while the upper approximation contains those elements of U which are related (up to ) to any element of X:

De nition 2.2.5 For a given (0; 1] and a fuzzy relation R in U characterized by 2 the membership function R : U U [0; 1]; the objects x and y in U will be called ! indiscernible if R(x; y) :

Proposition 2.2.6 Let (U; R) be a fuzzy approximation space and 1; 2 (0; 1] be 2 such that 1 2:Then R (X) R (X) and R (X) R (X): 1 2 2 1 c Proof. For any x R 1 (X); R(x; y) < 1 for all y X : But since 1 2;we 2 c 2 have R(x; y) < 2 for all y X : Thus, x R (X) showing that R (X) R (X): 2 2 2 1 2 Similarly, if x R (X); then R(x; y) 2 1 for some y X depicting that 2 2 2 x R (X): Hence, R (X) R (X): 2 1 2 1 2. Fuzzi ed Rough Sets 22 2.3 Proposed Algorithm

Taking (U; R) as fuzzy approximation space, A as set of attributes and T; the information table, following algorithm is proposed to approximate subsets of U: As already mentioned, it is assumed that no two objects have the same attribute values and the attribute values may be qualitative or quantitative.

Algorithm 2.3.1 Step 1: Input the triplet (U; A; T ): Step 2: If the attribute values in the information table are de ned by linguistic terms instead of numerical values; assign a fuzzy membership function to the attribute values. Corresponding to each attribute value, choose the point where the membership is maximum. If the attribute values are numerical, move to step 3. Step 3: For each pair of objects x; y U; nd fuzzy relation R in U U by the 2 membership function n

R(x; y) = R(xi; yi)=n (1) i=1 X where n is the number of attributes and for any ith attribute (i = 1; 2; :::n); xi; yi are the corresponding fuzzy values of the objects x and y; respectively. Here, R is a fuzzy relation in [0; 1] [0; 1] for the fuzzi ed attribute values characterized by the membership function

(xi; yi) = 1 xi yi for all i = 1; 2; :::; n: (2) R j j Step 4: Input the set of objects X U to be approximated. Step 5: Input degree of relationship (0; 1]: 2 Step 6: Calculate lower and upper approximations of X as de ned in De nition 2.2.2.

Step 2 assigns grades to attribute values of the decision matrix in Step 1 from

[0; 1]. For a pair of fuzzi ed attribute values (xi; yi) for an ith attribute corresponding to the pair of objects (x; y), the fuzzy relation R is a similarity relation which gives the degree to which both the values are related, that is, it measures similarity between them while the relation R averages these degrees for each pair of objects. Step 3 calculates this relation R. In steps 4 and 5, we input (crisp) set to be approximated (which may be a decision partition class) and the degree to which the elements are to be kept similar. The last step calculates approximations to signify the certain and probable members. 2. Fuzzi ed Rough Sets 23

Fuzzification Finding similarity Finding lower and of data set relation upper approximations

Fig. 2.1. Algorithm for nding approximations using alpha-fuzzi ed rough sets

Remark 2.3.2 R; being a re exive and symmetric relation, is a fuzzy compatibility relation which measures compatibility among the elements. Intuitively, compatibility or proximity relations are used to model the vague concept of `likeness' or `sameness'. Ovchinnikov [44] clari ed the idea behind proximity relation as follows. Suppose X is a nite set of objects and A is a nite set of their attributes such that each object a X has at least one attribute p A. Let X(p) be the set of all a X with attribute 2 2 2 p. We say that an object a resembles an object b if and only if they belong to the same subset X(p) for some attribute p A. This idea can be formalized as follows. Let R 2 be a binary relation on X de ned by aRb if there is p A such that a; b X(p). Then 2 2 R is a re exive symmetric binary relation. But, when there are three or more values of attribute p; fuzzy binary relations are indispensable to give relation among objects. Therefore, the need of fuzzy relation R in Algorithm 2.3.1 to nd the resemblance or compatibility among the objects subject to the given attributes is apparent.

2.3.1 Illustrative Example

Given a set of cost contingency techniques U = u; v; w; x; y; z = pre-determined f g f percentage, expert judgement, risk analysis, regression analysis, Delphi technique and PERT, NASA's joint con dence level model ; a set of attributes A = complexity, g f team e ort, accuracy, risk concerned and an information table given below,we are g aiming to nd the approximations of a subset X = u; v; x of U using Algorithm f g 2.3.1. Step 1: Input the triplet (U; A; T ) given by Table 2:2: Step 2: The semantics of the given linguistic terms (qualitative attribute values) 2. Fuzzi ed Rough Sets 24

u v w x y z complexity low low medium high medium high team e ort low low medium low high low accuracy low medium high medium high medium risk concerned low medium high medium high high

Table 2.2: Information Table Attribute/Objects u v w x y z complexity 0.75 0.75 0.5 0.25 0.5 0.25 team e ort 0.25 0.25 0.5 0.25 0.75 0.25 accuracy 0.25 0.5 0.75 0.5 0.75 0.5 risk concerned 0.75 0.5 0.25 0.5 0.25 0.25

Table 2.3: Fuzzi ed Information Table are given by triangular fuzzy membership functions as in g. 1 and g. 2 of g. 2.2.

low medium high high medium low

0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1

Fig. 1. Semantics of three terms for Fig. 2. Semantics of three terms for the the set {team effort, accuracy} set {complexity, risk concerned}

Fig. 2.2. Fuzzy membership functions

We notice that regarding the attributes "complexity" and "risk concerned", "low" is the best value while "high" is the worst value. In the rest of the attributes "low" and "high" have exactly the opposite meaning. Thus, we have assigned two di erent membership functions for the linguistic term sets complexity, risk concerned and f g team e ort, accuracy : The points where these terms have their maximum member- f g ship value are shown in Table 2.3. Step 3: Evaluate fuzzy relation as de ned in Step-3 of Algorithm 2.3.1. The resulting fuzzy relation R in U U is given in Table 2:4. Step 4: Input the set X = u; v; x of objects which is to be approximated. f g Step 5: Input degree of relationship = 0:9375: Step 6: Calculate lower and upper approximations of X = u; v; x using De ni- f g tion 2.2.2 as below: Here Xc = w; y; z . From Table 2.4 it can be seen that u and v are the only f g elements whose degree of relationship with respect to R to all the elements in Xc, 2. Fuzzi ed Rough Sets 25

R u v w x y z u 1 0.875 0.625 0.75 0.5625 0.6875 v 0.875 1 0.75 0.875 0.6875 0.8125 w 0.625 0.75 1 0.75 0.9375 0.8125 x 0.75 0.875 0.75 1 0.6875 0.9375 y 0.5625 0.6875 0.9375 0.6875 1 0.75 z 0.6875 0.8125 0.8125 0.9375 0.75 1

Table 2.4: Fuzzy relation among objects (contingency techniques) that is, w; y; z is less than = 0:9375: Therefore, R (X) = u; v : While the upper f g approximation contains all those elements whose degree of relationship with at least one element of X is 0:9375 or more. Since R is re exive fuzzy relation, elements in X will be in its upper approximation. Among the rest of objects (w; y; z); z is the only object such that R(z; x) = 0:9375 = for the element x in X and so is in R (X). It is clear from the above table that w and y are related to all elements of X with degree less than 0:9375. Thus, R (X) = u; v; x; z : f g We conclude that the techniques u (pre-determined percentage) and v (expert judgement) can be classi ed as members of X with respect to R up to a classi cation degree = 0:9375: While x (regression analysis) and z (NASA's joint con dence level model) can be classi ed as possible members of X with respect to R up to a classi cation degree = 0:9375:

Remark 2.3.3 The purpose of introducing degree of relationship is to give a margin to the data sets in which no two objects has exactly the same attribute values. Instead of taking them to be exactly similar; we are considering the second strongest degree to which two elements might be related. If it still does not work, we'll choose the third strongest degree and so on. Therefore, in Example 2.3.1 the degree is chosen to be 0.9375 which is the second strongest degree to which two elements can be related.

2.4 Properties of fuzzi ed rough approximations

Lemma 2.4.1 For any x; y U; the membership function R(x; y) of fuzzy relation 2 R de ned in equation 1 satis es the following property:

R(x; y) = 1 if and only if x = y: 2. Fuzzi ed Rough Sets 26

Proof. If x = y for any x; y U; then 2 n R(x; x) = R(xi; xi)=n i=1 n P = (1 xi xi )=n i=1 j j n = P(1)=n = n=n = 1: i=1 P Conversely, if R(x; y) = 1 then

n R(xi; yi)=n = 1 i=1 n P = (1 xi yi ) = n ) i=1 j j n P = n xi yi = n ) i=1 j j n P = xi yi = 0 ) i=1 j j = Pxi yi = 0 for all i=1,2,...,n. ) j j = xi = yi for all i=1,2,...,n. ) = x = y (Since no two objects have the same attributes). )

Following properties of the fuzzi ed lower and upper rough approximations hold.

Theorem 2.4.2 Let (U; R) be fuzzy approximation space, (0; 1]: Then for X U 2 we have

(1) R (X) X R (X);

(2) R ( ) = = R ( ); ; ; ;

(3) R (U) = U = R (U);

c c (4) R (X ) = (R (X)) ;

c c (5) R (X ) = (R (X)) ;

(6) X Y = R (X) R (Y ) and R (X) R (Y ); )

(7) R1 R2 = R2 (X) R1 (X) and R1 (X) R2 (X); ) (8) R (X Y ) = R (X) R (Y ); \ \ (9) R (X Y ) R (X) R (Y ); [ [ 2. Fuzzi ed Rough Sets 27

(10) R (X Y ) = R (X) R (Y ); [ [

(11) R (X Y ) R (X) R (Y ): \ \ Proof. (1) (3) Straightforward. (4) For any x U; 2 c x R (X ) 2 c c R(x; y) < for all y (X ) = X () 2 R(x; y) for any y X () 6 2 c x R (X) x (R (X)) : () 62 () 2 (5) For any x U; 2 c x R (X ) 2 c R(x; y) for some y X () 2c R(x; y) < for all y X () 6 2 x R (X) x (R (X))c: () 62 () 2 c c c (6) For any x R (X); R(x; y) < for all y X : In particular, since Y X ; 2 c 2 we may write R(x; y) < for all y Y which implies that x R (Y ): Thus, 2 2 R (X) R (Y ): Also, if x R (X); then R(x; y) for some y X Y which implies that 2 2 x R (Y ): Hence, R (X) R (Y ): 2 (7) The proof follows from De nition 2.2.2 and the fact that R1 R2 implies R (x; y) R (x; y) for all x; y U: 1 2 2 (8) By using part (6) above and the fact that X Y X;Y we may write \ R (X Y ) R (X);R (Y ) \ and so, R (X Y ) R (X) R (Y ): \ \

For the reverse inclusion, let x R (X) R (Y ): Then, x R (X) and x R (Y ), 2 c \ 2 c 2 that is, R(x; y) < for all y X and R(x; z) < for all z Y which gives c c2 c 2 R(x; u) < for all u X Y = (X Y ) and so x R (X Y ): Thus, R (X) 2 [ \ 2 \ \ R (Y ) R (X Y ) and hence R (X) R (Y ) = R (X Y ): \ \ \ (9) Since X;Y X Y , by part (6) of this theorem, we may write [ R (X);R (Y ) R (X Y ) [ and so, R (X) R (Y ) R (X Y ): [ [ (10) As X;Y X Y; by using part (6) we get [

R (X); R (Y ) R (X Y ) [ and so, R (X) R (Y ) R (X Y ): [ [ 2. Fuzzi ed Rough Sets 28

For the reverse inclusion, let x R (X Y ): Then 2 [

R(x; y) for some y X Y 2 [ The above condition can be expressed as

R(x; y) for some y X 2 or R(x; y) for some y Y 2

This gives us that x R (X) or x R (Y ), that is, x R (X) R (Y ): Hence, 2 2 2 [ R (X Y ) = R (X) R (Y ): [ [ (11) From part (6) above and the fact that X Y X;Y we get \

R (X Y ) R (X); R (Y ) \ and so, R (X Y ) R (X) R (Y ): \ \

Remark 2.4.3 (1) If = 1 then R (X) = X = R (X). Indeed, from part 1 of the

above theorem we have R (X) X R (X) for any X U and for any x X; c 2 from Lemma 2.4.1 we have R(x; z) = 1 for all z X , that is, R(x; z) < 6 2 (= 1) for all z Xc which implies X R (X) and so X = R (X): Also, for 2 any x R (X); R(x; y) (= 1) for some y X: But since R(x; y) [0; 1] 2 2 2 we get R(x; y) = 1 which implies, from Lemma 2.4.1, that x = y: Thus, x X 2 as y X which implies R (X) X: Hence, R (X) = X: 2 (2) The reverse inclusions in parts (8) and (10) do not hold which can be justi ed by Example 2.3.1. If we take X = u; v; w and Y = v; w; x and = 0:625: f g f g Then, by De nition 2.2.2, we get R (X) = ;R (Y ) = and since X Y = f g f g [ u; v; w; x ; we have R (X Y ) = u : Thus, R (X Y ) R (X) R (Y ): Also, f g [ f g [ 6 [ R (X) = u; v; w; x; y; z and R (Y ) = u; v; w; x; y; z . And since X Y = f g f g \ v; w , we have R (X Y ) = v; w; y; z which shows that R (X) R (Y ) f g \ f g \ 6 R (X Y ): \ c c (3) Since U = R (X) (R (X) R (X)) R (X); and the three sets (R (X) ; (R (X) [ [ R (X));R (X) )are disjoint, they form partition of U. These three sets are the negative, boundary and positive regions respectively.

2.5 Fuzzi ed rough membership function

In crisp set theory, an element either belongs to or does not belong to a set. Thus membership of an element to a crisp set can be represented through a characteristic 2. Fuzzi ed Rough Sets 29 function which maps an element to 1 if it belongs to the set and to 0 if it does not belong to that set. In rough set theory, the notion of membership function is generalized and is interpreted as a degree of certainty to which an element belongs to the set. Following the same concept, fuzzi ed rough membership function can be de ned. De nition 2.5.1 For a given fuzzy approximation space (U; R) and (0; 1]; we 2 de ne a relation in U U as below:

xy if and only if R(x; y) :

Then class for any x U will be [x] = y U : xy : 2 f 2 g This relation is a compatibility relation and each class is a compatibility class. Using this relation we can de ne fuzzi ed rough membership function [x] X R : U [0; 1] for a crisp set X asR (x) = j \ j for all x U: X X [x] ! j j 2 Example 2.5.2 In Example 2.2.3, for = 0:9; classes are given below: [x1] = x1; x3 = [x3]; [x2] = x2 , [x4] = x4; x5 = [x5]; [x6] = x6 : f g f g f g f g Using these classes, the membership of elements of U with X will be as below:

R R X (x1) = x1; x3 = x1; x3 = 1 = X (x3); R jf gj jf gj X (x2) = = x2 = 0; R j;j jf gj R X (x4) = x5 = x4; x5 = 1=2 = X (x5); R jf gj jf gj (x6) = = x6 = 0: X j;j jf gj Remark 2.5.3 fuzzi ed rough membership function and classes can be used to de ne the approximations as below:

R (1) R (X) = x U : (x) = 1 = x U :[x] X : Indeed, f 2 X g f 2 g

x R (X) 2 c R(x; y) < for all y X () 2 x y for any y Xc () 6 2 [x] X [x] X = [x] () () \ R (x) = 1: () X

R (2) R (X) = x U : (x) > 0 = x U :[x] X = ? : Indeed, f 2 X g f 2 \ 6 g

x R (X) 2 R(x; y) for some y X () 2 xy for some y X () 2 [x] X = ? [x] X > 0 () \ 6 () j \ j R (x) > 0: () X 2. Fuzzi ed Rough Sets 30

R (3) Clearly, BNDR(X) = R (X) R (X) = x U : 0 < (x) < 1 : f 2 X g Proposition 2.5.4 The fuzzi ed rough membership function has the following properties.

(1) R (x) = 1 if and only if x R (X); X 2

R c (2) (x) = 0 if and only if x (R (X)) ; X 2

R (3) 0 < (x) < 1 if and only if x BR (X); X 2

R R (4) c (x) = 1 (x) for any x U; X X 2

R R R (5) X Y (x) max(X (x); Y (x)) for any x U; [ 2 R R R (6) X Y (x) min(X (x); Y (x)) for any x U: \ 2 Proof. Straightforward.

2.6 Measures associated with fuzzi ed rough sets This section presents some measures associated with fuzzi ed rough sets. The main concern is to measure completeness of knowledge provided by a given tolerance fuzzy relation. Pawlak [47] presented the accuracy and roughness measures associated with rough set approximations. The accuracy measure is the ratio of the lower approximation to the upper approximation while the roughness measure is complement of the accuracy measure. The purpose of introducing this measure is to capture the degree of completeness of our knowledge about the set X or to express quality of an approximation. The roughness measure as complement of accuracy measure is interpreted as degree of incompleteness. As a generalization of these measures, we introduce an accuracy measure using fuzzi ed rough approximations as follows. De nition 2.6.1 Let (U; R) be a fuzzy approximation space and (0; 1]. The 2 accuracy measure of a subset X of U by R is de ned as follows.

R (X) (X;R) = j j A R (X)

where X = and X denotes the cardinality of the set X. 6 ; j j

Theorem 2.6.2 Let U be a universe set and R;R1;R2 be fuzzy relations over U. For

= X U; the accuracy measure (X;R) satis es the following properties. ; 6 A 2. Fuzzi ed Rough Sets 31

(1) (X;R) = 1 if and only if R (X) = R (X); A

(2) (X;R) = 0 if and only if R (X) = ; A ;

(3) For a xed R (X); (X;R) strictly monotonically increases with R (X) ; A j j

(4) For a xed R (X) = ; (X;R) strictly monotonically decreases with R (X) ; 6 ; A j j

(5) If 1 2; then (X;R) (X;R); A 1 A 2

(6) If R1 R2; then (X;R2) (X;R1): A A Proof. Straightforward.

Proposition 2.6.3 If R is an equivalence relation, then 1 fuzzi ed rough approximations degenerate into rough approximations.

Proof. Straightforward. Gediga and D•untsch [21] proposed a measure for the precision of approximation of X U for a given partition; which is not e ected by the approximation of X: For a given equivalence relation R, it is denoted and de ned as below.

apr (X) R (X;R) =

X j j In fuzzi ed rough environment it can be extended as follows. R (X) (X;R) = j j X j j Clearly, (X;R) (X;R): It may be noted that this measure requires complete A knowledge of X; whereas does not. It can be interpreted as the relative number of A elements in X which can be approximated by R.

Proposition 2.6.4 If 1 2; then (X;R) (X;R): 1 2 Proof. Straightforward. Yao [82] pointed out that the term `accuracy' should be de ned and interpreted accurately as it might be misleading in some cases. He revised some of the properties of the accuracy measure given by Pawlak [46, 47] and presented another measure of the `completeness' of knowledge, or the accuracy of approximations. The suggested measure is as follows.

POSR(X) + NEGR(X) (X;R) = j j Uj j apr (X) +j (japr (X))c R j R j = j j U apr (X) +j aprj (Xc) R R = j j Uj j j j 2. Fuzzi ed Rough Sets 32

Following his approach, we can extend this measure using fuzzy approximation space as below.

De nition 2.6.5 Let (U; R) be a fuzzy approximation space and (0; 1]. The 2 measure of quality of approximation of a subset X of U by R is de ned as follows.

R (X) + R (Xc) (X;R) = j j j j U j j The corresponding roughness measure is

(X;R) = 1 (X;R) R (X) R (X) = j U j j j

The measure can be interpreted as the measure of quality of approximation of the X generated partition X;Xc by R. f g

Theorem 2.6.6 For a given fuzzy approximation space (U; R) and X U; (X;R) satis es the following properties.

(1) (X;R) = 0 if and only if (R (X) = ; R (X) = U), ;

(2) For a xed R (X)(may be empty or non-empty); (X;R) strictly monotonically decreases with R (X) :

Proof. Straightforward.

The measure can serve well to check the quality of the X generated partition only, that is, X;Xc . To deal with decision partitions that contains more than two f g decision classes, we have the generalized measure of quality of approximation as follows.

R (Y ) : Y U=D (D;R) = fj j 2 g U P j j

Proposition 2.6.7 If D1 D2 (that is, for any D1(x) D1; there exists D2(x) D2 2 2 such that D1(x) D2(x)); where Di represents the partition induced by the decision b b b b b b attribute Di, then (D1;R) (D2;R): b b b Proof. Straightforward.

Example 2.6.8 Consider a fuzzy relation R over a universe set U = x1; x2; x3; f x4 as de ned in Table 2.5. Taking = 0:7; the lower approximations, the upper g approximations and the values of measures ; ; for all non-empty subsets X of A U have been shown in Table 2.6. 2. Fuzzi ed Rough Sets 33

R x1 x2 x3 x4 x1 1 0.3 0.9 0.6 x2 0.3 1 0.4 0.7 x3 0.9 0.4 1 0.5 x4 0.6 0.7 0.5 1

Table 2.5: A tolerance fuzzy relation

X R (X) R0:7(X) 0:7 A x1 x1; x3 0 0 1/2 f g ; f g x2 x2; x4 0 0 1/2 f g ; f g x3 x1; x3 0 0 1/2 f g ; f g x4 x2; x4 0 0 1/2 f g ; f g x1; x2 x1; x2; x3; x4 0 0 0 f g ; f g x1; x3 x1; x3 x1; x3 1 1 1 f g f g f g x1; x4 x1; x2; x3; x4 0 0 0 f g ; f g x2; x3 x1; x2; x3; x4 0 0 0 f g ; f g x2; x4 x2; x4 x2; x4 1 1 1 f g f g f g x3; x4 x1; x2; x3; x4 0 0 0 f g ; f g x2; x3; x4 x2; x4 x1; x2; x3; x4 1/2 2/3 1/2 f g f g f g x1; x2; x3 x1; x3 x1; x2; x3; x4 1/2 2/3 1/2 f g f g f g x1; x2; x4 x2; x4 x1; x2; x3; x4 1/2 2/3 1/2 f g f g f g x1; x3; x4 x1; x3 x1; x2; x3; x4 1/2 2/3 1/2 f g f g f g x1; x2; x3; x4 x1; x2; x3; x4 x1; x2; x3; x4 1 1 1 f g f g f g Table 2.6: All non-empty subsets of U with their accuracy measures 2. Fuzzi ed Rough Sets 34

t-norms 1) standard min operator M (x; y) = min x; y ; (the largest t-norm [28]) T f g 2) algebraic product P (x; y) = xy; 3) the bold intersectionT (also calledLukasiewicz t-norm) L(x; y) = max 0; x + y 1 ; T f g t-conorms 1) standard max operator M (x; y) = max x; y ; S f g 2) the probabilistic sum P (x; y) = x + y xy; S 3) bounded sum L(x; y) = min 1; x + y : S f g Table 2.7: Some famous t-norms and t-conorm

2.7 Generalization of fuzzi ed rough sets using logical connectives

In this section, we shall de ne an extension of fuzzi ed rough sets by using some of the logical connectives, particularly, implication and triangular norms. We'll start this section by recalling some basic de nitions of logical connectives. A triangular norm, or shortly t-norm, [27] is an increasing, associative and commutative mapping : [0; 1] [0; 1] [0; 1] satisfying (1; x) = x for all x [0; 1]: T ! T 2 A triangular conorm, or shortly t-conorm, is any increasing, commutative and associative mapping : [0; 1] [0; 1] [0; 1] satisfying (0; x) = x for all x [0; 1]: S ! S 2 A negation operator is a decreasing mapping : [0; 1] [0; 1] satisfying N ! (0) = 1 and (1) = 0: Standard negation operator (or a negator) denoted by s N N N is de ned as s(x) = 1 x for all x [0; 1]:A negator is called involutive if N 2 N ( (x)) = x for all x [0; 1]: N N 2 An implication operator (or implicator) is any mapping : [0; 1] [0; 1] I ! [0; 1] satisfying (1; 0) = 0 and (1; 1) = (0; 1) = (0; 0) = 1: It is called border I I I I implication if it satis es the condition that (1; x) = x for all x [0; 1]: I 2 Based on a t-norm ; t-conorm and an involutive negator ; an implicator T S N I de ned as (x; y) = ( (x); y) for all x; y [0; 1] is called an S implicator. I S N 2 Another implicator based on a t-norm de ned as (x; y) = sup [0; 1] : I T I f 2 (x; y) ; provided that is continuos, is called an R-implicator (residual im- T g T plicator). Some famous t-norms, t-conorms, negations and implicators have been mentioned in Tables 2.7 and 2.8. For further details about implication operators, we refer to [55].

Theorem 2.7.1 [52] Every S implicator and R implicator is a border implicator. Proof. Straightforward. 2. Fuzzi ed Rough Sets 35

S implicators 1)Lukasiewicz implicator L(x; y) = min 1; 1 x + y I f g based on L and s; S N 2) Kleene-Dienes implicator KD(x; y) = max 1 x; y I f g based on M and s; S N 3) Kleene-Dienes-Lukasiewicz implicator P (x; y) = 1 x + xy IS based on P and s: S N R implicators 1) theLukasiewicz implicator L based on L; I T 2) the G•odelimplicator G(x; y) = 1 for x y and I G(x; y) = y elsewhere, based on M ; 3)I the Gaines implicator (x; y)T = 1 for x y and I4 (x; y) = y=x elsewhere, based on P : I4 T Table 2.8: Some famous implicators

By using Remark 2.5.3, we can re-write lower and upper fuzzi ed rough approximations as

R (X) = x U : ( for all y U) xy = y X f 2 2 ) 2 g (3) and R (X) = x U : ( there exist) xy y X f 2 ^ 2 g where = and are an implicator and a triangular norm, which have been used to ) ^ interpret the conditions [x] X and [x] X = ? respectively. \ 6 Radzikowska and Kerre [52] introduced a broad class of fuzzy rough sets which they called ( ; ) fuzzy rough sets, when represents implicator and ; a triangular I T I T norm. The de nition is stated below.

De nition 2.7.2 Let FAS = (X;R) be a fuzzy approximation space and and be I T a border implicator and a t-norm, respectively. An ( ; ) fuzzy rough approximation ; I T in FAS is a mapping AprI T : (X) (X) (X) de ned for every A (X) FAS F ! F F 2 F by ; AprI T (A) = (F (A); F T (A)); FAS AS AS I where FAS (A) and FAST (A)) are de ned for every x X as: I 2

FAS (A)(x) = inf (R(x; y);A(y)); y XI I 2 FAST (A)(x) = sup (R(x; y);A(y)): y XT 2

The fuzzy set FAS (A) (respectively FAST (A)) is called an lower (respectively I I upper) fuzzy rough approximation of A in FAS: T 2. Fuzzi ed Rough Sets 36

From equations 3, it is clear that the lower and upper fuzzi ed rough approximations mean:

(x) = 1 if and only if for all y U (R(x; y) = X (y) = 1) R (X) 2 ) (4) (x) = 1 if and only if for all y U (R(x; y) X (y) = 1) R (X) 2 ^ As pointed out by Radzikowska and Kerre [52], = and can be interpreted as ) ^ an implicator and a t-norm ; respectively. With this interpretation, we get the I T following extended form of fuzzi ed rough approximations in De nition 2.2.2. De nition 2.7.3 Let (U) be the collection of all fuzzy sets over the nite universe F U; R : U U [0; 1] be the fuzzy relation and (0; 1]: Let a fuzzy relation ! 2 R : U U [0; 1] be de ned as ! R(x; y) if R(x; y) R (x; y) = ( 0 if R(x; y) <

Then lower and upper ( ; ) fuzzi ed rough approximations are fuzzy sets de- I T ned for any f (U) as 2 F

R (f)(x) = inf (R (x; y); f(y)) y UI 2 and R (f)(x) = sup (R (x; y); f(y)) y UT 2 where is a border implicator and is a t-norm. The pair (R (f); R (f)) will be I T called ( ; ) fuzzi ed rough set. I T

Example 2.7.4 Consider the fuzzy relation R over a universe set U = x1; x2; x3; f x4 as de ned in Table 2.5 in Example 2.6.8 and a fuzzy set f de ned below. g

0.6 if x = x1 8 0.9 if x = x2 f(x) = > > 0.2 if x = x3 <> 0.5 if x = x4 > Using De nition 2.7.3, and taking> = 0:5; R is calculated as in Table 2.9. :> Using the same de nition and taking G and M as the implicator and the T-norm I T (de ned in Tables 2.7 and 2.8), the lower and upper approximations of the fuzzy set f as calculated in below.

R (f)(x1) = inf G(R0:5(x1; x1); f(x1)); G(R0:5(x1; x2); f(x2)); 0:5 fI I G(R0:5(x1; x3); f(x3)); G(R0:5(x1; x4); f(x4)) I I g = inf G(1; 0:6); G(0; 0:9); G(0:9; 0:2); G(0:6; 0:5) fI I I I g = inf 0:6; 1; 0:2; 0:5 = 0:2: f g 2. Fuzzi ed Rough Sets 37

R0:5 x1 x2 x3 x4 x1 1 0 0.9 0.6 x2 0 1 0 0.7 x3 0.9 0 1 0.5 x4 0.6 0.7 0.5 1

Table 2.9: Fuzzy relation R-0.5

R0:5(f)(x1) = sup M (R0:5(x1; x1); f(x1)); M (R0:5(x1; x2); f(x2)); fT T M (R0:5(x1; x3); f(x3)); M (R0:5(x1; x4); f(x4)) T T g = sup M (1; 0:6); M (0; 0:9); M (0:9; 0:2); M (0:6; 0:5) fT T T T g = sup 0:6; 0; 0:2; 0:5 = 0:6: f g Similarly, the rest of the calculations can be made that are summarized as below.

0.2 if x = x1 8 0.5 if x = x2 R0:5(f)(x) = > > 0.2 if x = x3 <> 0.2 if x = x4 > > :> 0.6 if x = x1 8 0.9 if x = x2 R0:5(f)(x) = > > 0.6 if x = x3 <> 0.7 if x = x4 > > 2.7.1 Proposed Algorithm :

The relation R has been used to ful l the requirement R(x; y) as in equation 4. Thus, two elements will be related if their degree of relationship is at least : Using these ( ; ) approximations, Algorithm 2.3.1 can be revised as follows. I T Algorithm 2.7.5 Step 1: Input the triplet (U; A; T ) where U is a nite universe, A is the set of attributes and T is the given information table. Step 2: Fuzzify the information table by assigning suitable values from the interval [0; 1] to each attribute value. Step 3: For each pair of objects x; y U; nd fuzzy relation R in U U by the 2 membership function n

R(x; y) = R(xi; yi)=n i=1 X where n is the number of attributes and for any ith attribute (i = 1; 2; :::n); xi; yi are the corresponding fuzzy values of the objects x and y; respectively. Here, R is 2. Fuzzi ed Rough Sets 38

R u v w x y z u 1 0.75 0 0.5 0 0 v 0.75 1 0.5 0.75 0 0.625 w 0 0.5 1 0.5 0.875 0.625 x 0.5 0.75 0.5 1 0 0.875 y 0 0 0.875 0 1 0.5 z 0 0.625 0.625 0.875 0.5 1

Table 2.10: A fuzzy relation R-alpha a fuzzy relation in [0; 1] [0; 1] for the fuzzi ed attribute values characterized by the membership function

(xi; yi) = 1 xi yi for all i = 1; 2; :::; n: R j j Step 4: Input degree of relationship (0; 1]: 2 Step 5: Find R : U U [0; 1] as in De nition 2.7.3. ! Step 6: Input a fuzzy set f over U which is to be approximated.

Step 7: Find ( ; ) lower and upper fuzzi ed rough approximations as in Def- I T inition 2.7.3 by using a suitable border implication and a t-norm : I T The above algorithm presents an expedient technique for approximations of fuzzy sets. The upper and lower approximations are fuzzy sets which can be determined by choosing a suitable implicator and a t-norm.

Example 2.7.6 Consider the information system in Example 2.3.1. Steps 1-3 are the same as in algorithm 2.3. Taking = 0:5; we nd R using table 2.4. The calculated values are shown in table 2.10. Suppose we want to approximate the fuzzy set f on U de ned below.

0.7 if s = u 8 0.4 if s = v > 0.3 if s = w > f(s) = > > 0.8 if s = x <> 0.1 if s = y > > 0.5 if s = z > > Using De nition 2.7.3, the Lukasiewicz> implicator L and the Lukasiewicz T-norm : I L (de ned in Tables 2.7 and 2.8), we nd the lower and upper 0:5 fuzzi ed approxi- T 2. Fuzzi ed Rough Sets 39 mations as below. 0.65 if s = u 8 0.4 if s = v > 0.225 if s = w > R (f)(s) = > > 0.625 if s = x <> 0.1 if s = y > > 0.5 if s = z > > :> 0.7 if s = u 8 0.55 if s = v > 0.30 if s = w > R (f)(s) = > > 0.80 if s = x <> 0.175 if s = y > > 0.675 if s = z > > > 2.8 Comparative analysis:

Rough set theory introduced by Pawlak [46] was based on crisp equivalence classes. It works well in discrete data sets. But de ning classes with sharp boundaries on data sets containing continuos values is impractical and sometimes impossible. This fact led to the development of fuzzy rough set theories. Pioneering work in this regard can be seen in [17]. Other noteworthy techniques include a generalized approach by Radzikowska and Kerre [53] and a further generalization of this work by Ouyang et al. [43]. Dubois and Prade [17] suggested two almost similar approaches to combine fuzzy and rough sets. First one is to use fuzzy similarity relation in place of crisp equivalence relation and in the second approach; fuzzy partition is introduced to get fuzzy granules. This approach proved its worth in de ning granules of objects with soft boundaries and thus making rough set theory more practical. Radzikowska and Kerre [53] presented another approach (( ; )-fuzzy rough sets) of combining the two theories I T by using fuzzy implicator and triangular norm. They categorized fuzzy rough sets in three classes based on S-, R- and QL- implicators and proved that fuzzy rough set by Dubois and Prade [17] is a particular case of the newly presented fuzzy rough sets. The above mentioned techniques of combining fuzzy and rough sets were based on fuzzy similarity relation that is analogous to equivalence relation in crisp case. These approaches are valid only if transitivity in fuzzy relation is considered as sup-min transitivity. In case of -transitivity (in which standard min operator is replaced by T -norms), these approaches are unsuccessful in producing disjoint clusters which are T expected in case of similarity relations. 2. Fuzzi ed Rough Sets 40

Fuzzy rough sets for -similarity relations (also called tolerance relations) were T discussed by Ouyang et al. [43] as a generalization of ( ; )-fuzzy rough sets. They I T proved some properties that were not satis ed by ( ; )-fuzzy rough sets. I T These hybrid models produced worthy approaches in combining fuzzy and rough sets but are de cient in two main aspects. Firstly, they do not provide a way to cluster objects by using fuzzy relations that are not (sup-min or ) transitive. Secondly, they T yield fuzzy sets as approximations whose interpretations are highly dependent on the choice of fuzzy relation. The induction of di erent fuzzy relations will highly e ect the membership values in lower and upper approximations as well as their interpretations.

For example, R1(x; y) = min(x; y) and R2(x; y) = 1 x y are both fuzzy relations j j but they di er a lot in their actual meanings. The former one gives minimum possible values while the latter gives nearness among the elements. Also, the same membership value may have di erent meanings with respect to di erent fuzzy relations. Like, in the above mentioned relations, R1(0:2; 0:4) = R2(0:1; 0:9) = 0:2 whose interpretation is totally di erent with respect to R1 and R2. Due to these reasons the existing theories are dicult to implement in practical situations. D'eer et al. [16] established a mathematical framework for the most relevant fuzzy rough set models proposed in the literature and critically evaluated them. They investigated and compared these models on the basis of their properties. They declared three main properties as the most important ones among the rest and discussed them in detail. These are stated in parts 1, 6 and 7 of Theorem 2.4.2. It is concluded in the said paper that more likely none of the existing models is able to retain all the properties of Pawlak's original model. Fuzzy variable precision rough set model presented by Zhao et al. [87] satis es most theoretical properties, yet it does not satisfy the inclusion property, that is, property 1 of Theorem 2.4.2. The other fuzzy rough set models that satisfy some of these properties, but not all, are proposed in [11, 13, 57, 58, 84]. In this chapter, we presented a technique to overcome these shortcomings. Approx- imations have been constructed with a view point of clustering objects with respect to fuzzy relations that may not be transitive, that is, compatibility relations. A control parameter has been introduced which allows to choose relational degree among the elements as close as we wish to without being exactly similar. Fuzzy relation R used in the presented algorithms measures compatibility among the elements. This fuzzy relation R and crisp lower and upper approximations make it easy to interpret and implement the results in practical situations. The approximation operators proposed in this paper satisfy all the properties marked as important in [16]. The inclusion property (Theorem 2.4.2, part 1) is intuitively expected to hold in any such structure. The set monotonicity property (Theorem 2.4.2, part 6) makes this technique possible and reliable to apply in classi cation problems. Furthermore, the relation monotonicity 2. Fuzzi ed Rough Sets 41

C1 C2 C3 C4 C5 C6 x1 0.8 0.1 0.1 0.5 0.2 0.3 x2 0.3 0.5 0.2 0.8 0.1 0.1 x3 0.2 0.2 0.6 0.7 0.3 0.2 x4 0.6 0.3 0.1 0.2 0.5 0.3 x5 0.3 0.4 0.3 0.3 0.6 0.1 x6 0.2 0.3 0.5 0.3 0.5 0.2 x7 0.3 0.3 0.4 0.2 0.6 0.2 x8 0.3 0.4 0.3 0.1 0.4 0.5 x9 0.3 0.2 0.5 0.4 0.4 0.2

Table 2.11: Samples of credit card evaluation problem property (Theorem 2.4.2, part 7) plays its role in applications like attribute reduction [16]. Thus, the presented model is more advantageous practically in the sense that it has the above mentioned signi cant properties. As a numerical experiment, we consider an evaluation problem of credit card applicants formulated by Tsang et al. [71] (Example 3.1, Table 1) and approximate two (decision partition) sets by Dubois and Prade [17] technique and by algorithm 2.3.1.

The problem consists of nine elements (applicants) x1; x2; x3; x4; x5; x6; x7; x8; x9 and the applicants are described by six fuzzy attributes C1(best education);C2(better education);C3(good education);C4(high salary);C5(middle salary);C6(low salary). The information system is shown in Table 2.11. The sets to be approximated are

A = x1; x2; x4; x7 and B = x3; x5; x6; x8; x9 : The approximations by the approach f g f g in [17] are calculated using two fuzzy relations R1 and R2 de ned below.

R1(xi; xj) = min(Ck(xi);Ck(xj)) k

R2(xi; xj) = min 1 Ck(xi) Ck(xj) k f j jg The resulting lower and upper approximations are given in Table 2.12. The approximations have also been calculated using algorithm 2.7.5 (taking M as t-norm T and KD as implicator) and the last column of this table shows approximations by I algorithm 2.3.1. The abrupt change in approximations (by changing fuzzy relation) can be noticed in C 1 and C 2 columns of the table. Depending upon R1 and R2, membership values of the approximations have di erent meanings and cannot be interpreted in the same manner. Since R2 is comparatively a better fuzzy relation to measure similarity among elements, we compare our results of algorithm 2.3.1 and algorithm 2.7.5 with C 2. FRS lower approximation of A contains three elements x1; x2 and x4 and it is evident from C 2 and C 3 of the table that membership of these three 2. Fuzzi ed Rough Sets 42

C 1 C 2 C 3 C 4 R1 R2 algo2( = 0:9) algo1( = 0:9) 0.9 x = x1 0.5 x = x1 1 x = x1 0.9 x = x2 0.4 x = x2 1 x = x2 8 8 8 AR > 0.9 x = x4 > 0.3 x = x4 > 1 x = x4 x1; x2; x4 > > > f g <> 0.8 x = x7 <> 0.1 x = x7 <> 0.07 x = x7 0 otherwise 0 otherwise 0 otherwise > > > > 0.2 x = x3 > 0.6 x = x3 > 0 x = x3 :> :> :> 0.1 x = x5 0.9 x = x5 0.93 x = x5 8 8 8 0.2 x = x6 0.9 x = x6 0.93 x = x6 x1; x2; x4; AR > > > f > 0.1 x = x8 > 0.7 x = x8 > 0 x = x8 x5; x7 > > > g < 0.2 x = x9 < 0.8 x = x9 < 0.9 x = x9 > 1 otherwise > 1 otherwise > 1 otherwise > > > > 0.8 x = x3 > > 1 x = x3 :> :> 0.4 x = x3 :> 0.9 x = x5 0.07 x = x5 8 0.1 x = x5 8 > 0.8 x = x6 8 > 0.07 x = x6 BR > 0.3 x = x8 > x3; x8; x9 > 0.9 x = x8 > > 1 x = x8 f g > > 0.1 x = x > < 0.8 x = x < 9 < 0.10 x = x 9 0 otherwise 9 > 0 otherwise > > 0 otherwise > > > > :> 0.5 x = x1 > :> 0.1 x = x1 :> 0 x = x1 0.6 x = x2 0.1 x = x2 8 0 x = x2 8 0.7 x = x4 8 x3; x5; x6; BR 0.1 x = x4 > 0 x = x4 f > > 0.9 x = x > x ; x ; x > 0.2 x = x > 6 > 0.93 x = x 7 8 9 < 7 < 0.9 x = x < 7 g 1 otherwise 7 1 otherwise > > 1 otherwise > > > > :> > :> Table 2.12::> Comparison of approximations 2. Fuzzi ed Rough Sets 43 elements (to lower approximation of A) is greater than rest of the elements. FRS upper approximation of A contains ve elements that have maximum (1) or 0:9 (2nd maximum) degree in C 2 and C 3: Similar interpretation holds for upper approximation of B: The elements in the lower approximation of B in C 4 have maximum or second maximum degree in C 3 (algorithm 2.7.5) but di er slightly in C 2. This di erence can be removed by appropriate choice of fuzzy relation in C 2. Although researchers have obtained remarkable achievements regarding fuzzy rough sets, yet there are a series of problems to be solved such as how fuzzy relation will be selected and how to interpret the results according to that relation. Without a satisfactory answer to these questions, the implementation of the existing fuzzy rough sets approaches to practical issues is questionable. In view of the above comparative results, the salient features of the proposed techniques can be listed as below; (1) The technique is compared with the state-of-the-art fuzzy rough set presented by Dubois and Prade [17] that uses a fuzzy similarity relation. The approximations by algorithm 1 and 2 use a fuzzy tolerance relation, yet there is no signi cant di erence among the results. (2) A given concept can be approximated either by two crisp sets (lower and upper approximations) using algorithm 1 or by fuzzy sets using algorithm 2 according to the situation. (3) The explicit tolerance fuzzy relation used makes the techniques easy to implement and interpret. (4) The proposed approximation operators satisfy the important properties that are necessary from the application perspective. Thus, it is hoped that the algorithms presented in this paper will be more e ective in handling issues like risk analysis, classi cation problems, attribute reduction and pattern recognition.

2.9 Conclusion

We introduced a general approach for fuzzi cation of rough sets through a tolerance fuzzy relation and also investigated some properties of the proposed technique and its membership function. An algorithm has been presented to show practicality of this new interesting approach. Generalization of this technique using logical connectives accompanied by an algorithm is also discussed. Comparison of this technique with other renowned ones shows its validity and ecacy. The proposed algorithms in this chapter dealt with a special fuzzy proximity (intransitive) relation. For an arbitrary fuzzy relation R; the approach presented in this chapter needs further investigations. Chapter 3

Generalized Hesitant Fuzzy Rough Sets

3.1 Introduction

As a generalization of fuzzy rough sets, the concept of generalized hesitant fuzzy rough sets (GHF RS) is presented in this chapter. It is an endeavor to de ne rough approximations of a collection of hesitant fuzzy sets over a given universe. To this end, elements of the universe are initially clustered using a set valued map and then hesitant fuzzy sets are aggregated by using lower and upper approximation operators. These operators produce hesitant fuzzy sets which aggregate hesitant fuzzy elements. Structural and topological properties associated with GHF RS have also been examined. The new model is further employed to design a three-way decision analysis technique which preserves many properties of classical techniques but needs less e ort and computation. Unlike the existing approaches, the alternatives can be clustered and selected jointly by using a set valued mapping. This feature makes its application area broader. Moreover, this method is applied to an example, where risk analysis issue is discussed for the selection of energy projects.

3.2 Approximations of Generalized hesitant fuzzy rough sets (GHF RS)

This section introduces a new technique for combining rough and hesitant fuzzy sets which involves no hesitant fuzzy relation. Its structural and topological properties have also been examined. In order to keep the things simple throughout this chapter, we shall consider the case of discrete structures only.

44 3. Generalized Hesitant Fuzzy Rough Sets 45

Several aggregation operators and measures have been de ned for hesitant fuzzy sets [74, 77, 78]. The main purpose in de ning these operators and measures is to average them or to nd distance or similarity between them. Instead of averaging hesitant fuzzy elements, one may be interested in nding their minimum and maximum. For example, an investment company may be interested in nding maximum chances of pro t or minimum chances of loss for investing a sum of money in a project subject to some given factors (which may be taken as hesitant fuzzy sets). Also, most of these existing measures are based on algebraic sum and algebraic product of hesitant fuzzy elements. In practical applications, like in decision making problems, these measures will change the actual values provided by experts. This may be misleading in some situations. To overcome these problems, we introduce two hesitant fuzzy operators which will be called lower and upper hesitant fuzzy rough approximation operators. These operators are based on a given collection of hesitant fuzzy sets over a universe set U and a set-valued map T which maps each element of U to a non-empty subset of U: This map generalizes the concept of equivalence classes in rough set theory and allows us to choose clusters of elements without any restriction. Moreover, the operators of hesitant fuzzy union Y and hesitant fuzzy intersection Z have been used to de ne the approximations so that the actual values will be retained.

3.2.1 (GHF RS) approximation operators and its properties

De nition 3.2.1 Let = hi : i I be a ( nite) collection of hesitant fuzzy sets H f 2 g on U; that is, HF (U) and T : U P (U) be a set valued mapping, where H ! P (U) = P (U) . The lower and upper approximations of with respect to T are n; H hesitant fuzzy sets T and T de ned for any x U as: H H 2

T (x) = Z Z hi x0 H i I x0 T (x) 2 2 and T (x) = Y Y hi x0 H i I x0 T (x) 2 2 respectively. The pair T ; T is called a generalized hesitant fuzzy rough set H H (GHFRS) with respect to T . is called hesitant fuzzy de nable if T = T . H H H Example 3.2.2 A student Z is planning to appear in a competitive exam. He has to choose three subjects among a list of optional subjects S1, S2, S3, S4, S5, S6 keeping in view their three attributes h1 (overall scoring), h2 (time for preparation) and h3

(length of syllabus). There is one more restriction. If he chooses subject S2 then he must choose S3 as well. He seeks advice of three experts and ask them to grade the subjects according to their attributes. These grades are given in Table 3.1 below: 3. Generalized Hesitant Fuzzy Rough Sets 46

h1 h2 h3 S1 (:2;:4;:3) :3;:5;:6 :6;:6;:7 f g f g S2 :8;:5;:6 :2;:4;:5 :7;:7;:5 f g f g f g S3 :1;:2;:5 :8;:7;:6 :4;:3;:7 f g f g f g S4 :4;:2;:3 :9;:6;:7 :6;:4;:3 f g f g f g S5 :6;:5;:8 :4;:4;:5 :3;:5;:7 f g f g f g S6 :3;:5;:5 :8;:5;:5 :3;:5;:6 f g f g f g Table 3.1: Hesitant fuzzy sets showing grades of student Z

Here = h1; h2; h3 is the collection of hesitant fuzzy sets and U = S1;S2;S3; H f g f S4;S5;S6 is the set of alternatives. According to the given restriction, we de ne the g mapping T : U P (U) as below: !

Si if i 1; 4; 5; 6 T (Si) = f g 2 f g ( S2;S3 if i 2; 3 f g 2 f g

According to De nition 3.2.1, the lower and upper approximations of the subjects S1 to S6 are as below:

T (S1) = :2;:4;:3 ; T (S1) = :6;:7 ; H f g H f g T (S2) = :2;:4;:5 ; T (S2) = :5;:6;:7;:8 ; H f g H f g T (S3) = :1;:2;:3;:4;:5 ; T (S3) = :8;:7;:6 ; H f g H f g T (S4) = :4;:2;:3 ; T (S4) = :9;:6;:7 ; H f g H f g T (S5) = :4;:5 ; T (S5) = :6;:5;:8;:7 ; H f g H f g T (S6) = :3;:5 ; T (S6) = :8;:5;:6 : H f g H f g The lower approximations give the minimum (expected) grading and upper approximations give the maximum (expected) grading. By comparing these values, he will choose

S4, S5 and S6 as they have relatively better grading than the rest.

Example 3.2.3 Suppose that a software company desires to hire a system analysis engineer. After preliminary screening, four candidates y1, y2; y3 and y4 quali ed for further evaluation. A committee of three decision-makers, has been formed to select the suitable candidate for the post according to the following ve criteria:

(1) emotional steadiness (h1),

(2) oral communication skill (h2),

(3) personality (h3),

(4) past experience (h4),

(5) self-con dence (h5). The decision makers provide their preferences in anonymity and the decision matrix is given in Table 3.2. 3. Generalized Hesitant Fuzzy Rough Sets 47

h1 h2 h3 h4 y1 (:2;:4;:7) (:3;:5;:6) (:6;:6;:7) (:5;:6;:8) y2 (:2;:4;:5) (:5;:7;:7) (:1;:2;:5) (:6;:7;:8) y3 (:3;:4;:7) (:4;:4;:6) (:6;:7;:7) (:4;:6;:9) y4 (:5;:6;:8) (:4;:4;:5) (:3;:5;:5) (:5;:5;:8)

Table 3.2: Hesitant fuzzy decision matrix

Here = h1; h2; h3; h4 is the collection of hesitant fuzzy sets and U = y1; y2; H f g f y3; y4 is the set of alternatives. To categorize the objects (candidates) into classes g according to the preferences given in Table 3.2; we de ne the image sets of T : U ! P (U) as below.

T (x) = y U : s(hi(y)) s(hi(x)) < 0:05 for all i = 1; 2; 3; 4 f 2 j j g where s(h(x)) denotes the score of h(x) de ned in De nition 1.3.6. Thus, the image sets of mapping T : U P (U) will be as follows: ! yi if i 2; 4 T (yi) = f g 2 f g ( y1; y3 if i 1; 3 f g 2 f g According to De nition 3.2.1, the lower and upper approximations of the candidates y1 to y4 are as below:

T (y1) = T (y3) = :2;:3;:4;; 5;:6 ; T (y1) = T (y3) = :6;:7;:8;:9 ; H H f g H H f g T (y2) = :1;:2;:4;:5 ; T (y2) = :6;:7;:8 ; H f g H f g T (y4) = :3;:4;:5 ; T (y4) = :5;:6;:8 : H f g H f g The lower approximations give the minimum (expected) grading and upper approximations give the maximum (expected) grading. By comparing these values, y1 and y3 have relatively the better grading than the rest.

Lemma 3.2.4 For any collection hi(xj): i I; j J of hesitant fuzzy elements f 2 2 g on U; Z hi (xj) Y hi (xj) : i I;j J i I;j J 2 2 2 2 Proof. For the given collection of hesitant fuzzy elements, we have

Z hi (xj) = inf hi(xj) ; Y hi (xj) = sup hi(xj) ; i I;j Jf g f g i I;j J ! 2 2 i I;j J ! i I;j J 2 2 + 2 2 + 2 2 + + Z hi (xj) = inf hi (xj) ; Y hi (xj) = sup hi (xj) ; i I;j J i I;j Jf g i I;j J i I;j Jf g 2 2 ! 2 2 2 2 ! 2 2 + + which gives Z hi (xj) Y hi (xj) and Z hi (xj) Y hi (xj) : i I;j J i I;j J i I;j J i I;j J 2 2 ! 2 2 ! 2 2 ! 2 2 ! Thus, by De nition 1.3.2, Z hi (xj) Y hi (xj) : i I;j J i I;j J 2 2 2 2 3. Generalized Hesitant Fuzzy Rough Sets 48

The above lemma states that for a given collection of hesitant fuzzy elements, their intersection is contained in their union. Thus, intersection and union of hesitant fuzzy elements generalize the concept of union and intersection for crisp sets.

Theorem 3.2.5 If T : U P (U) is a set valued mapping, then for any ! H HF (U);T T : H H Proof. Follows directly from the above lemma and De nition 1.3.3.

Proposition 3.2.6 For any hesitant fuzzy elements h1(u); h2(v); h3(w) and h4(x) on

U, if h1(u) h2(v) and h3(w) h4(x) then h1(u) h3(w) h2(u) h4(x): Z Z Proof. For the given hesitant fuzzy elements we have

+ + h1(u) h2(v) = h1(u) h2(v); h1 (u) h2 (v); ) + + h3(w) h4(x) = h(w) h(x); h (w) h (x): ) 3 4 3 4

Now, (h1(u) h3(w)) = inf h(u); h(w) h(u) h(v) and (h1(u) Z f 1 3 g 1 2 Z h3(w)) = inf h(u); h(w) h(w) h(x) indicating that (h1(u) h3(w)) f 1 3 g 3 4 Z is lower bound of h2(v); h4(x) : Thus, (h1(u) Z h3(w)) inf h2(v); h4(x) = f g + + f g (h2(v) h4(x)) : Similarly, (h1(u) h3(w)) (h2(v) h4(x)) : The above arguments Z Z Z and De nition 1.3.2, show that h1(u) h3(w) h2(v) h4(x): Z Z The above proposition can be generalized to any number of hesitant fuzzy sets which may be formally stated as below.

Corollary 3.2.7 Let hi : i I and hj : j J be two collections of hesitant fuzzy f 2 g f 2 g sets on U: If, for any y U; hi(y) hj(y) for all i I; j J; then Z hi (y) 2 2 2 i I 2 Z hj (y) : j J 2 Proof. Straightforward. Following the similar arguments, results in Proposition 3.2.6 and Corollary 3.2.7 can also be proved for hesitant fuzzy union. Using De nition 3.2.1 and operations mentioned in De nition 1.3.3, it is natural to investigate the properties of upper and lower generalized hesitant fuzzy approximation operators. The next theorem expresses the details.

Theorem 3.2.8 Let T : U P (U) be a set valued mapping. Then for all ; 1; 2 ! H H H 3. Generalized Hesitant Fuzzy Rough Sets 49

HF (U) and for any x U; 2 c (HFL1) T (x) = T c (x ) ; H Hc c = hi : hi H f 2 Hgc (HFU1) T H(x) = (T c (x)) ; H (HFL2) T 1 2 (x) = T 1 (x) Z T 2 (x); H eH H H 1 2 = h1 h2 : h1 1; h2 2 ; H e H f e 2 H 2 H g (HFU2) T 1 2 (x) = T 1 (x) Y T 2 (x); H dH H H 1 2 = h1 h2 : h1 1; h2 2 ; H d H f d 2 H 2 H g (HFL3) 1 2 T 1 (x) T 2 (x); H H ) H H 1 2 means hi hj for all hi 1, hj 2; H H 2 H 2 H (HFU3) 1 2 T 1 (x) T 2 (x); H H ) H H (HFL4) T 1 (x) Y T 2 (x) = T 1 2 (x); H H H dH (HFU4) T 1 2 (x) = T 1 (x) Z T 2 (x): H eH H H

Proof. (HFL1) Using part 6 of Theorem 1.3.5, we get c c c T c (x ) = Y Y hi (x0) H x0 T (x) i I ! 2 2 c c = Z Y hi (x0) x T (x) i I 02 2 = Z Z hi (x0) = T (x) : x T (x) i I H 02 2

(HFU1) Using part 6 of Theorem 1.3.5, we get c c c (T c (x)) = Z Z hi (x0) H x0 T (x) i I ! 2 2 c c = Y Z hi (x0) x T (x) i I 02 2 = Y Y hi (x0) = T (x) : x T (x) i I H 02 2

(HFL2) Using parts 2 and 3 of Theorem 1.3.5 and taking 1 = hi : i I and H f 2 g 2 = hj : j J ; we may write H f 2 g

T (x) = (h h )(x ) 1e 2 Z Z i e j 0 H H x T (x) i I;j J 02 2 2 !

= Z ( Z hi(x0)) Z ( Z hj (x0)) x T (x) i I j J 02 2 2 !

= Z Z hi(x0) Z Z Z hj(x0) x T (x) i I x T (x) j J " 02 2 # " 02 2 !#

= T 1 (x) Z T 2 (x): H H 3. Generalized Hesitant Fuzzy Rough Sets 50

(HFU2) Using parts 2 and 3 of Theorem 1.3.5 and taking 1 = hi : i I and H f 2 g 2 = hj : j J ; we may write H f 2 g

T 1d 2 (x) = Y Y (hi d hj)(x0) H H x T (x) i I;j J 02 2 2 !

= Y ( Y hi(x0)) Y ( Y hj (x0)) x T (x) i I j J 02 2 2 !

= Y Y hi(x0) Y Y Y hj(x0) x T (x) i I x T (x) j J " 02 2 # " 02 2 !# = T 1 (x) Y T 2 (x): H H

(HFL3) Using Corollary 3.2.7, we have

T 1 (x) = Z Z hi x0 Z Z hj x0 = T 2 (x): H i I j J H x0 T (x) x0 T (x) ! 2 2 2 2 (HFU3) Using Corollary 3.2.7, we have

T 1 (x) = Y Y hi x0 Y Y hj x0 = T 2 (x): H i I j J H x0 T (x) x0 T (x) ! 2 2 2 2 (HFL4) Taking 1 = hi : i I and 2 = hj : j J and by making use of H f 2 g H f 2 g part 4 of Theorem 1.3.5, we have

T (x) = (h h )(x ) 1d 2 Z Z i d j 0 H H x T (x) i I;j J 02 2 2 !

= Z Z hi(x0) Y Z hj (x0) x T (x) i I j J 02 " 2 2 !#

= Z Z hi(x0) Y Z Z hj(x0) x T (x) i I x T (x) j J " 02 2 # " 02 2 !#

= T 1 (x) Y T 2 (x): H H

(HFU4) Taking 1 = hi : i I and 2 = hj : j J and by making use of H f 2 g H f 2 g part 4 of Theorem 1.3.5, we have

T 1e 2 (x) = Y Y (hi e hj)(x0) H H x T (x) i I;j J 02 2 2 !

= Y Y hi(x0) Z Y hj (x0) x T (x) i I j J 02 " 2 2 !#

= Y Y hi(x0) Z Y Y hj(x0) x T (x) i I x T (x) j J " 02 2 # " 02 2 !# = T 1 (x) Z T 2 (x): H H 3. Generalized Hesitant Fuzzy Rough Sets 51

The above results have been proved for hesitant fuzzy elements. Since the element has been chosen arbitrarily, these results also hold for hesitant fuzzy sets.

Corollary 3.2.9 With usual notations, the following results hold:

c c (HFL10 ) T = T c ; (HFU10 ) T = (T c ) ; H H H H (HFL20 ) T 1 2 = T 1 e T 2 ; (HFU20 ) T 1d 2 = T 1 d T 2 ; H eH H H H H H H (HFL30 ) 1 2 T 1 T 2 ; (HFU30 ) 1 2 T 1 T 2 ; H H ) H H H H ) H H (HFL40 ) T 1 d T 2 = T 1 2 ; (HFU40 ) T 1e 2 = T 1 e T 2 : H H H dH H H H H Proof. Straightforward.

Properties HFL10 and HFU10 show that the GHF R approximation operators T H and T are dual to each other. Properties HFL20 ;HFU20 ;HFL40 and HFU40 show that H both the lower and upper GHF R approximation operators T and T are distributive H H with respect to both the union and intersection of hesitant fuzzy sets.

Corollary 3.2.10 The properties (HFL20 ); (HFL40 ); (HFU20 ) and (HFU40 ) in Corol- lary 3.2.9 can be generalized to any number of collections of hesitant fuzzy sets, that is, T = T ;T = T , T = T and T = T where e i e i d i d i d i d i e i e i i IH i I H i IH i I H i IH i I H i IH i I H 2 2 2 2 2 2 2 2 i HF (U) for each i I . H 2 Proof. Straightforward.

3.2.2 Singleton GHF R approximation operators

In some cases it may be necessary to approximate a single hesitant fuzzy set instead of a collection of hesitant fuzzy sets. In that case = hi : i reduces to a H f 2 Ig singleton set containing the hesitant fuzzy set to be approximated only. We shall then call the approximation operators as singleton GHF R approximation operators. It is very obvious that the singleton GHF R operators will satisfy all the properties that are satis ed by the GHF R approximation operators. Besides these properties, these singleton approximation operators undergo some special properties that may not be true or may not even be discussed in the former case. Thus there is a need to study these operators separately.

De nition 3.2.11 Let h be a hesitant fuzzy set on U; that is, h HF (U) and T : 2 U P (U) be a set valued mapping, where P (U) = P (U) . The lower and upper ! n; 3. Generalized Hesitant Fuzzy Rough Sets 52 singleton approximations of h with respect to T are hesitant fuzzy sets T (h) and T (h) de ned for any x U as: 2 T (h)(x) = Z h x0 x0 T (x) 2 and T (h)(x) = Y h x0 x0 T (x) 2 respectively. The pair T (h); T (h) is called a singleton generalized hesitant fuzzy rough (SGHF R) set with respect to T . h is called hesitant fuzzy de nable if T (h) = T (h).

Example 3.2.12 We consider a universe set U = a; b; c; d and a hesitant fuzzy set f g h over U given by: h(a) = :2;:5;:6 ; h(b) = :3;:4 ; h(c) = :3;:5;:7;:8 ; h(d) = :3;:4;:8 : Let the f g f g f g f g set-valued map T : U P (U) be de ned as below: ! T (a) = a; b; c; d ;T (b) = b; c; d ;T (c) = c; d ;T (d) = d : f g f g f g f g We rst nd the lower and upper SGHF R approximations of h using De nition 3.2.11 as follows:

T (h)(a) = Z h(x) x T (a) 2 = h(a) Z h(b) Z h(c) Z h(d) = :2;:5;:6 :3;:4 :3;:5;:7;:8 :3;:4;:8 f g Z f g Z f g Z f g = :2;:3;:4 f g T (h)(a) = Y h(x) x T (a) 2 = h(a) Y h(b) Y h(c) Y h(d) = :2;:5;:6 :3;:4 :3;:5;:7;:8 :3;:4;:8 f g Y f g Y f g Y f g = :3;:4;:5;:6;:7;:8 f g Similarly, we nd rest of the approximations as below:

T (h)(b) = :3;:4 ; T (h)(b) = :3;:4;:5;:7;:8 ; f g f g T (h)(c) = :3;:4;:5;:7;:8 ; T (h)(c) = :3;:4;:5;:7;:8 ; f g f g T (h)(d) = :3;:4;:8 ; T (h)(d) = :3;:4;:8 : f g f g It may be noticed that T (h)(c) = T (h)(c) = h(c): Thus, approximations of a de n- 6 able hesitant fuzzy set may not be equal to the hesitant fuzzy set itself.

Lemma 3.2.13 Let T : U P (U) be a set valued mapping: Then T (hU ) = hU = ! T (hU ) and T (h0) = h0 = T (h0). 3. Generalized Hesitant Fuzzy Rough Sets 53

Proof. Straightforward.

Lemma 3.2.14 If for any x1; x2 U; T (x1) T (x2) then T (h)(x2) T (h)(x1) and 2 T (h)(x1) T (h)(x2): Proof. Follows directly from the result in part 2 of Remark 1.3.4. In GHF R sets, the mapping T : U P (U) plays a vital role. By imposing ! certain conditions on T : U P (U), GHF R approximations satisfy some extra ! properties that help in constructing GHF R set models.

A re exive binary relation is a set-valued mapping from U to P (U) such that x is an element of T (x). Similarly, symmetric, transitive and Euclidean binary relations can also be expressed as set-valued mappings. In this context, we shall refer to T : U P (U) as ! re exive if for all x U; x T (x) 2 2 symmetric if for all x; y U;if x T (y);then y T (x) 2 2 2 transitive if for all x; y U; y T (x);then T (y) T (x) 2 2 Euclidean if for all x; y U; y T (x);then T (x) T (y) 2 2 With respect to these special types of set-valued mappings, the approximation operators have additional properties.

Theorem 3.2.15 If T : U P (U) is a set valued mapping, then for any h ! 2 HF (U) the following results hold:

(1) If T : U P (U) is re exive, then T (h) h T (h): ! (2) If T : U P (U) is symmetric, then h T (T (h)): ! (3) If T : U P (U) is transitive, then ! (i) T (h) T (T (h)); (ii) T (T (h)) T (h): (4) If T : U P (U) is Euclidean, then ! (i) T (T (h)) T (h); (ii) T (h) T (T (h)): Proof. (1) For any x U; since T is re exive, x T (x) for all x U; using 2 2 2 Remark 1.3.4, we have T (h)(x) = Z h(x0) h(x) Y h(x0) = T (h)(x): x T (x) x T (x) 02 02 3. Generalized Hesitant Fuzzy Rough Sets 54

(2) For any x U; 2

T (T (h))(x) = Z (T (h)(x0)) = Z (Yh(x00)): x T (x) x T (x) x T (x ) 02 02 002 0 As T : U P (U) is symmetric, for any x T (x) we have x T (x ): This implies ! 0 2 2 0 that h(x) Y h(x00) for all x0 T (x) x T (x ) 2 002 0 which gives h(x) Z ( Y h(x00)) x T (x) x T (x ) 02 002 0 = h(x) T (T (h))(x) for all x U ) 2 Hence h T (T (h)): (3)(i) For any x U; 2

T (T (h))(x) = Z T (h)(xi) = Z ( Z h(x00)) xi T (x) xi T (x) x T (xi) 2 2 002

As T : U P (U) is transitive; xi T (x) and x T (xi) implies x T (x); that ! 2 00 2 00 2 is, T (xi) T (x): This together with Lemma 3.2.14 gives

Z h(x0) Z h(x00) for all xi T (x): x T (x) x T (xi) 2 02 002 By Proposition 3.2.6 and Theorem 1.3.5, we have

Z h(x0) Z ( Z h(x00)) x T (x) xi T (x) x T (xi) 02 2 002 which means T (h)(x) T (T (h))(x) for all x U and the result follows. 2 (3)(ii) For any x U; 2

T (T (h))(x) = Y (T (h)(x0)) = Y ( Y h(x00)): x T (x) x T (x) x T (x ) 02 02 002 0 By transitivity of T, T (x ) T (x) for all x U: This together with Lemma 3.2.14 0 0 2 gives Y h(x00) Y h(x0) for all x0 T (x) x T (x ) x T (x) 2 002 0 02 By Proposition 3.2.6 and part 1 of Theorem 1.3.5, we have

Y ( Y h(x00)) Y h(x0) x T (x) x T (x ) x T (x) 02 002 0 02 This implies that T (T (h))(x) T (h)(x) for all x U. Thus the result follows. 2 3. Generalized Hesitant Fuzzy Rough Sets 55

(4)(i) Since T : U P (U) is a Euclidean mapping, for any x U we have ! 2 T (T (h))(x) = Y ( Z h(x00)) x T (x) x T (x ) 02 002 0 Y ( Z h(x00)) as T (x) T (x0) for all x0 T (x) x T (x) x T (x) 2 02 002 = Z h(x0) x T (x) 02 = T (h)(x)

Thus T (T (h)) = T (h): (4)(ii) For any x U; 2 T (T (h))(x) = Z (T (h)(x0)) = Z ( Y h(x00)): x T (x) x T (x) x T (x ) 02 02 002 0 As T : U P (U) is Euclidean, T (x) T (x ) for all x T (x) which gives ! 0 0 2 Y h(x0) Y h(x00) for all x0 T (x) x T (x) x T (x ) 2 02 002 0

= Y h(x0) Z ( Y h(x00)) ) x T (x) x T (x) x T (x ) 02 02 002 0 = T (h)(x) T (T (h))(x) for all x U ) 2 = T (h) T (T (h)) ) Thus, T (T (h)) = T (h): The following example shows that the properties in the above theorem may not hold if the conditions of T being re exive, symmetric, transitive or Euclidean are not satis ed.

Example 3.2.16 Consider a universe set U = a; b; c; d and a hesitant fuzzy set h f g over U and a set-valued map T : U P (U) de ned as: ! h(a) = :2;:5;:9 ; h(b) = :3;:4 ; h(c) = :3;:5;:7;:8 ; h(d) = :3;:4;:8 ; f g f g f g f g T (a) = b; c; d ;T (b) = c; d ;T (c) = a ;T (d) = a; c : f g f g f g f g The upper and lower SGHF R approximations of h with respect to T are given below: T (h)(a) = :3;:4 ; T (h)(a) = :3;:4;:5;:7;:8 ; f g f g T (h)(b) = :3;:4;:5;:7;:8 ; T (h)(b) = :3;:4;:5;:7;:8 ; f g f g T (h)(c) = :2;:5;:9 ; T (h)(c) = :2;:5;:9 ; f g f g T (h)(d) = :2;:3;:5;:7;:8 ; T (h)(d) = :3;:5;:7;:8;:9 : f g f g The map T is not re exive as for example a = T (a): Also, 2 T (h)(a) = :3;:4 f g h(a) = :2;:5;:9 f g T (h)(a) = :3;:4;:5;:7;:8 f g 3. Generalized Hesitant Fuzzy Rough Sets 56

+ + as T (h)(a) = :3 h(a) = :2 and h (a) = :9 T (h) (a) = :8: This shows that the property T (h) h T (h) may not hold if the map T is not re exive. The map T is not symmetric as b T (a) but a = T (b): Also, 2 2 h(b) = :3;:4 f g T (T (h))(b) = Z T (h)(x) x T (b) 2 = T (h)(c) Z T (h)(d) = :2;:5;:9 :3;:5;:7;:8;:9 f g Z f g = :2;:3;:5;:7;:8;:9 f g as h (b) = :3 T (T (h)) (b) = :2: Thus, the property h T (T (h)) may not hold if the set-valued map T is not symmetric. The map T is not transitive as c T (a) but T (c) = a * T (a) = b; c; d : Also, 2 f g f g T (h)(a) = :3;:4 f g T (T (h))(a) = Z T (h)(x) x T (a) 2 = T (h)(b) Z T (h)(c) Z T (h)(d) = :3;:4;:5;:7;:8 :2;:5;:9 :2;:3;:5;:7;:8 f g Z f g Z f g = :2;:3;:4;:5;:7;:8 f g as T (h)(a) = :3 T (T (h))(a) = :2: Thus, the property T (h) T (T (h)) may not hold if the map T is not transitive. In addition,

T (T (h))(c) = Y T (h)(x) x T (c) 2 = T (h)(a) = :3;:4;:5;:7;:8 f g T (h)(c) = :2;:5;:9 f g as T (T (h))(c) = :3 T (h)(c) = :2: Thus, the property T (T (h)) T (h) may not hold if the map T is not transitive. The map T is not Euclidean as c T (a) but T (a) = b; c; d * T (c) = a : Also, 2 f g f g T (T (h))(a) = Y T (h)(x) x T (a) 2 = T (h)(b) Y T (h)(c) Y T (h)(d) = :3;:4;:5;:7;:8 :2;:5;:9 :2;:3;:5;:7;:8 f g Y f g Y f g = :3;:4;:5;:7;:8;:9 f g T (h)(a) = :3;:4 f g 3. Generalized Hesitant Fuzzy Rough Sets 57

as T (T (h))+(a) = :9 T (h)+(a) = :4: Thus, the property T (T (h)) T (h) may not hold if the map T is not Euclidean. In addition,

T (h)(a) = :3;:4;:5;:7;:8 f g T (T (h))(a) = Z T (h)(x) x T (a) 2 = T (h)(b) Z T (h)(c) Z T (h)(d) = :3;:4;:5;:7;:8 :2;:5;:9 :3;:5;:7;:8;:9 f g Z f g Z f g = :2;:3;:4;:5;:7;:8 f g

as T (h)(a) = :3 T (T (h))(a) = :2: Thus, the property T (h) T (T (h)) may not hold if the map T is not Euclidean.

3.3 Topological structure associated with GHF R sets and singleton GHF R sets

The relationship between topological spaces and rough and fuzzy rough approximation operators is well-known and is studied by many researchers. Kondo studied topological structures in rough sets [29]. Chang introduced the concept of fuzzy topological spaces and generalized many notions and concepts of topology in terms of fuzzy topology [9]. Hutton and Reilly gave separation properties for fuzzy topological spaces as the generalizations of standard topological notions [23]. By the generalization of fuzzy sets to hesitant fuzzy sets, the concept of fuzzy topology can further be generalized to hesitant fuzzy topology as below. In this section, we de ne hesitant fuzzy topological spaces and study some topological properties of GHF RS model. It is shown that union of de nable collections of hesitant fuzzy sets over U forms a hesitant fuzzy topology. Properties of this topology depend not only upon the hesitant fuzzy sets but also on the set valued map T: We start by de ning some de nitions related to hesitant fuzzy topology.

De nition 3.3.1 A hesitant fuzzy topology on a set U is a collection T of hesitant fuzzy sets on U satisfying:

(1) h0; hU T; 2 (2) if h1 and h2 belong to T; then so does h1 e h2; (3) for a given collection hi : i I of hesitant fuzzy sets on U; f 2 g if hi belongs to T for each i I; then so does d hi: 2 i I 2 3. Generalized Hesitant Fuzzy Rough Sets 58

If T is a hesitant fuzzy topology on U, then the pair (U; T) is called a hesitant fuzzy topological space. All those hesitant fuzzy sets which belong to T are called open hesitant fuzzy sets whereas complement of an open hesitant fuzzy sets is called closed hesitant fuzzy sets.

The collection h0; hU will be called an indiscrete hesitant fuzzy topology f g on U while the collection of all hesitant fuzzy sets over U will be called a discrete hesitant fuzzy topology. The following example illustrates the concept of a hesitant fuzzy topology.

Example 3.3.2 Let U = 1; 2; 3 be a universe set. Consider the collection T = f g h0; hU ; h1 of hesitant fuzzy sets where, h1 is de ned as below: f g h1(1) = 0; 0:2; 0:4 ; h1(2) = 0:1; 0:3 ; h1(3) = 0:5; 0:8; 0:9 . f g f g f g Clearly, h0 e hU = h0 e h1 = h0; h0 d hU = hU d h1 = h0 d hU d h1 = hU ; h0 d h1 = hU e h1 = h0 e hU e h1 = h1: Thus, T becomes a hesitant fuzzy topology on U. The open hesitant fuzzy sets of T are h0; hU and h1while closed hesitant fuzzy sets of T are c c h0; hU and h1; where h1 is given by: hc (1) = :6;:8; 1 ; hc (2) = :7;:9 ; hc (3) = :1;:2;:5 : 1 f g 1 f g 1 f g De nition 3.3.3 A map : HF (U) HF (U) is referred to as a hesitant fuzzy ! interior operator if and only if for all h1; h2 HF (U) it satis es: 2

1. (h1) h1 for all h1 HF (U); 2

2. (h1 h2) = (h1) (h2) for all h1; h2 HF (U); e e 2

3. ((h1)) = (h1) for all h1; h2 HF (U): 2 De nition 3.3.4 A map : HF (U) HF (U) is referred to as a hesitant fuzzy ! closure operator if and only if for all h1; h2 HF (U) it satis es: 2

1. h1 (h1) for all h1 HF (U); 2

2. (h1 h2) = (h1) (h2) for all h1; h2 HF (U); d d 2

3. ( (h1)) = (h1) for all h1 HF (U): 2 There exists an intimate relationship between structures based on rough sets and topological spaces. It is well known that if the underlying indiscernilbility relation is 3. Generalized Hesitant Fuzzy Rough Sets 59 an equivalence relation, tolerance relation or a pre-order or the set is a generalized rough set, then the collection of de nable sets give rise to a topology. (see [1, 29]). Here we consider a more general situation and we have the following result.

Theorem 3.3.5 Let T : U P (U) be a set valued map. Then the collection ! TF = h HF (U): T = T = HF (U): T = T is a hesitant 2 H H H H H H fuzzy topology on U. S

Proof. By Lemma 3.2.13, h0; hU TF . Now let h1; h2 TF . Then, there will 2 2 exist 1, 2 HF (U) such that T 1 = T 1 and T 2 = T 2 . Using (HFL20 ) and H H H H H H (HFU40 ) of Corollary 3.2.9, we may write

T 1 2 = T 1 e T 2 = T 1 e T 2 = T 1e 2 : H eH H H H H H H

Hence, h1 h2 TF since h1 h2 1 2: e 2 e 2 H e H Now let hi i I be a collection of members of TF . Then, there exists a collection f g 2 i : i HF (U) i I such that T i = T i for all i I. Corollary 3.2.10 implies that fH H g 2 H H 2 T = T = T = T . Hence T = T . And since h ; d i d i d i d i d i d i d i d i i IH i I H i I H i IH i IH i IH i I 2 i IH 2 2 2 2 2 2 2 2 we have d hi TF . Therefore, TF is a hesitant fuzzy topology on U: i I 2 2

Theorem 3.3.6 In TF ; every open hesitant fuzzy set is closed.

Proof. Straightforward.

In de ning hesitant fuzzy topology TF , the map T : U P (U) plays a vital role. ! By imposing certain conditions on T one can see change of behavior of open sets in

TF . This depicts its diversity in TF . In the subsequent results, some of its features have been highlighted.

Theorem 3.3.7 Let A be some xed ( nite) subset of [0; 1] and = hi : hi(x) = A H f for all i I : Then T = T = hi for all i I; that is, T (x) = T (x) = A. 2 g H H 2 H H Proof. Directly follow from the de nition of GHF RS. Converse of this result does not hold in general, as can be seen from the following example:

Example 3.3.8 Let U = 1; 2; :::; 10 and = h where h is a hesitant fuzzy set xf x a g H f g on U de ned as h(x) = for all x U: Then h is not a constant hesitant a[=1f 9 g 2 fuzzy set. Let T : U P (U) be de ned by T (x) = x for all x U. Then ! f g 2 T (x) = Z h (x0) = h(x) = Y h (x0) = T (x): This shows that the equality H x T (x) x T (x) H 02 02 T = T = h (h ) may hold even if h is a non-constant hesitant fuzzy set. H H 2 H 2 H 3. Generalized Hesitant Fuzzy Rough Sets 60

Theorem 3.3.7 and Example 3.3.8, highlight two important aspects of the hesitant fuzzy topology TF on U; rstly, any collection of constant hesitant fuzzy sets on U will always belong to TF for any set valued map T : U P (U) and secondly, TF not ! only contains constant hesitant fuzzy sets but also it may contain the non-constant hesitant fuzzy sets on U. This leads us to the following result.

Corollary 3.3.9 TF can never be an indiscrete hesitant fuzzy topology.

Proof. Straightforward. By imposing certain conditions on the set-valued mapping T : U P (U) and ! using singleton GHF R approximation operators T (h) and T (h), some additional hesitant fuzzy topologies emerge. These topologies are investigated in the rest of the section.

Theorem 3.3.10 If T (x) = and T (x) = U; then every h F is a x\U 6 ; x[U 2 H constant hesitant fuzzy2 set. 2

Proof. Since T (x) = ; there exists some y T (x); that is, y T (x) for x\U 6 ; 2x \U 2 all x U: This gives2 2 2 T (x) = Z ( Z hi(x0)) hi(y) Y ( Y hi(x0)) = T (x) H x T (x) i I x T (x) i I H 02 2 02 2 for any hi : But then, for any h F and x U we have 2 H 2 H 2 T (x) = T (x) = h(y): H H

Now, for any z U such that z = y; z T (x1) for some x1 U; since T (x) = U: 2 6 2 2 x[U 2 So, for any h F we have 2 H

h(z) = T (x1) = T (x1) = h(y) H H which shows that h is constant hesitant fuzzy set.

Theorem 3.3.11 If T (x) = a for all x U; where a is a xed element of U, then f g 2 T = h HF (U): T (h) = T (h) is a discrete hesitant fuzzy topology. F0 f 2 g Proof. For any h HF (U) and x U; consider 2 2 T (h)(x) = Z h(x0) = Z h(x0) = h(a) = Y h(x0) = Y h(x0) = T (h)(x): x T (x) x a x a x T (x) 02 02f g 02f g 02 This implies that h T for every h HF (U); which shows that T is a discrete 2 F0 2 F0 hesitant fuzzy topology. Since T (h) = T (h) does not imply T (h) = T (h) = h; the collections of hesitant fuzzy sets satisfying T (h) = h (respectively T (h) = h) will be di erent from TF : In the following it is proved that these collections also form hesitant fuzzy topologies. 3. Generalized Hesitant Fuzzy Rough Sets 61

Theorem 3.3.12 1 = h HF (U): T (h) = h is a hesitant fuzzy topology on f 2 g HF (U):

Proof. By Lemma 3.2.13, h0; hU 1: Also, by Corollary 3.2.9, we get 2

T (h1 h2) = h1 h2 for all h1; h2 1; e e 2 that is, h1 h2 1: e 2 Consider a sub-collection hi 1 : i I of 1: For any x U; from Corollary f 2 2 g 2 3.2.10, we have

T ( hi) = T (hi) = hi idI idI idI 2 2 2 so that d hi 1: Hence, 1 is a hesitant fuzzy topology on HF (U). i I 2 2

Theorem 3.3.13 2 = h HF (U): T (h) = h is a hesitant fuzzy topology on f 2 g HF (U):

Proof. By Lemma 3.2.13, h0; hU 2: Also, by Corollary 3.2.9, we get 2

T (h1 h2) = h1 h2 for all h1; h2 2; e e 2 that is, h1 h2 2: e 2 Consider a sub-collection hi 2 : i I of 2: For any x U; from Corollary f 2 2 g 2 3.2.10, we have

T ( hi) = T (hi) = hi idI idI idI 2 2 2 so that d hi 2: Hence, 2 is a hesitant fuzzy topology on HF (U). i I 2 2

Theorem 3.3.14 3 = T (h): h HF (U) is a hesitant fuzzy topology on HF (U). f 2 g

Proof. Using Lemma 3.2.13, h0 = T (h0) and hU = T (hU ) that is, h0; hU 3: 2 For any T (h1);T (h2) 3; from Corollary 3.2.9, we get T (h1) T (h2) = T (h1 h2) 2 e e 2 3 as h1 h2 HF (U): e 2 Also, for any sub-collection T (hi): i I of 3; from Corollary 3.2.10, we have f 2 g

d T (hi) = T ( d hi) 3 as d hi HF (U): i I i I 2 i I 2 2 2 2

Thus, 3 is a hesitant fuzzy topology on HF (U):

Theorem 3.3.15 4 = T (h): h HF (U) is a hesitant fuzzy topology on HF (U). f 2 g 3. Generalized Hesitant Fuzzy Rough Sets 62

Proof. Using Lemma 3.2.13, h0 = T (h0) and hU = T (hU ) that is, h0; hU 4: 2 For any T (h1); T (h2) 4; from Corollary 3.2.9, we get T (h1) T (h2) = T (h1 h2) 2 e e 2 4 as h1 h2 HF (U): e 2 Also, for any sub-collection T (hi): i I of 4; from Corollary 3.2.10, we have f 2 g d T (hi) = T ( d hi) 4 as d hi HF (U): i I i I 2 i I 2 2 2 2 Thus, 4 is a hesitant fuzzy topology on HF (U): It is well known that an interior operator (in crisp case) de nes a topology that I contains the xed points of . In case of rough sets, this interior operator is equivalent I to the lower rough approximation operator [83]. The result also holds in the case of fuzzy interior operator [75]. Generalization of this result using hesitant fuzzy interior operator leads to the following theorem.

Theorem 3.3.16 Taking T : U P (U) as a set-valued mapping and h HF (U), ! 2 the following are equivalent:

1. T : U P (U) is a re exive and transitive map; ! 2. The singleton upper GHF R approximation operator = T (h): HF (U) ! HF (U) is a hesitant fuzzy closure operator;

3. The singleton lower GHF R approximation operator = T (h): HF (U) ! HF (U) is a hesitant fuzzy interior operator.

Proof. Follows directly from Theorem 3.2.15 and Corollary 3.2.9. This theorem shows that upper and lower singleton GHF R operators become hesitant fuzzy closure and interior operators respectively when the underlying set- valued map T is a re exive and transitive map. The condition of re exivity and transitivity of T is necessary as the following example illustrates.

Example 3.3.17 Let U = a; b; c; d be a universe set and hesitant fuzzy set h be f g de ned as below: h(a) = :2;:5;:9 ; h(b) = :3;:4 ; h(c) = :3;:5;:7;:8 ; h(d) = :3;:4;:8 : We take f g f g f g f g set-valued map T : U P (U) as follows: ! T (a) = b; c; d ;T (b) = c; d ;T (c) = a ;T (d) = a; c : f g f g f g f g This map T is not re exive, as a = T (a) and it is not transitive as a T (c) but 2 2 T (a) * T (c): Also, T (h)(a) = Z h(x) x T (a) 2 = :3;:4 f g h(a) = :2;:5;:9 f g 3. Generalized Hesitant Fuzzy Rough Sets 63

as T (h) (a) = :3 h (a) = :2: Thus, the lower approximation operator is not an interior operator as it does not satisfy the rst axiom of De nition 3.3.3. Also,

h(a) = :2;:5;:9 f g T (h)(a) = Y h(x) x T (a) 2 = :3;:4;:5;:7;:8 f g + as h+(a) = :9 T (h) (a) = :8: This shows that the upper approximation operator is not a closure operator as it does not satisfy the rst axiom of De nition 3.3.4.

It is obvious from Theorem 3.3.16 that the pair (i(h); cl(h)); where i and cl are respectively the interior and closure operators, forms a SGHF R set. The hesitant fuzzy set h corresponding to the SGHF R set (i(h); cl(h)) = (T (h); T (h)) may not be unique as the following example shows.

Example 3.3.18 Let U = a; b be a universe set. Consider two hesitant fuzzy sets f g h1 and h2 on U and a set-valued map T : U P (U) as below: ! h1(a) = :2;:4;:8 ; h1(b) = :3;:7 ; h2(a) = :2;:7;:8 ; h2(b) = :3;:4;:7 and f g f g f g f g T (a) = T (b) = U:

The lower and upper SGHFR approximations of h1 and h2 are as below:

T (h1)(a) = :2;:3;:4;:7 = T (h1)(b); f g T (h2)(a) = :2;:3;:4;:7 = T (h2)(b); f g T (h1)(a) = :3;:4;:7;:8 = T (h1)(b); f g T (h2)(a) = :3;:4;:7;:8 = T (h2)(b): f g

Although h1 = h2; yet (T (h1); T (h1)) = (T (h2); T (h2)). 6 Theorem 3.3.19 Given that T : U P (U) is a re exive and transitive map, there ! exists a hesitant fuzzy topology on U such that = T (h): HF (U) HF (U) and ! = T (h): HF (U) HF (U) are the hesitant fuzzy interior and closure operators ! respectively.

Proof. Follows from Theorem 3.3.16.

3.4 Three way decision making with GHF RS model

This section presents three-way decision making methodology using GHF RS model. A risk analysis example has been presented to further illustrate the technique and comparison has been made with other notable techniques. 3. Generalized Hesitant Fuzzy Rough Sets 64

3.4.1 Proposed algorithm

GHF RS deduce three-way decisions using lower and upper approximations as explained below. If U is a given set of alternatives and T : U P (U) is set-valued map, then ! for each x U; P OS(x) represents its lower approximation, NEG(x) represents com- 2 plement of the upper approximation and BND(x) represents di erence of upper and lower approximations subject to the given collection of hesitant fuzzy sets on U: Here `complement' and `di erence' are the operations stated in De nition 1.3.3. Various steps involved in risk decision making method are illustrated below:

Algorithm 3.4.1 Step 1: For a given risk decision-making problem with a set of n alternatives U = A1;A2; :::; An and a set of m risk factors = P1;P2; :::; Pm ; f g H f g utilize the evaluations of decision makers for the risk factors involved in the given alternatives and establish a hesitant fuzzy decision matrix. Step 2: Taking into consideration the constraint, according to which some alternatives must be chosen together, de ne the map T : U P (U) such that T (Ai) = Aj : ! f Aj = Ai or Aj must be chosen with Ai for all i; j 1; 2; :::; n : g 2 f g Step 3: Evaluate POS(Ai);NEG(Ai) and BND(Ai) for each Ai (i = 1; 2; :::; n) c wherePOS(Ai) = T (Ai) ;NEG(Ai) = (T (Ai)) ;BND(Ai) = T (Ai) T (Ai) : H H H H Step 4: Calculate score for each risk factor estimated in Step 3 and average the values for each region using arithmetic mean. Step 5: Categorize each entry in the three regions as favorable (F ) or not favorable (NF ) according to the following rules:

F1 : For POS(Ai): If POS(Ai) Average(POS); decide Ai as favorable (F ); otherwise not favorable (NF );

F2 : For NEG(Ai): If NEG(Ai) Average(NEG); decide Ai as favorable (F ); otherwise not favorable (NF );

F3 : For BND(Ai): If BND(Ai) Average(BND); decide Ai as favorable (F ); otherwise not favorable (NF ); classify the alternatives into the best suitable region according to the following rules and obtain their corresponding values:

R1 : If only one of the values POS(Ai);NEG(Ai);BND(Ai) is favorable, then

Ai will be classi ed into that particular region;

R2 : If two or all three values POS(Ai);NEG(Ai);BND(Ai) are favorable, choose the one whose distance from its average value is maximum;

R3 : Ties will be broken arbitrarily if the respective distances are same. 3. Generalized Hesitant Fuzzy Rough Sets 65

Step 6: Rank the alternatives using their corresponding values according to the preference of the decision maker. Step 7: End.

Establish a hesitant fuzzy decision matrix Calculate POS(Ai), Calculate scores NEG(Ai) for HFEs and BND(Ai) Choose a suitable for each Ai set•valued map T

Categorize alternatives Find average of as favorable or not scores for each favorable region

Classify into POS region Rank Classify into according NEG region to the preference relation Classify into BND region

Form decision rules

Fig 3.1. A risk analysis algorithm with the aid of GHF RS

The above algorithm gives a technique to handle risk decision making problems. Here the map T : U P (U) clusters those alternatives which must be selected ! together. In Steps 3 and 4, scores of positive, negative and boundary regions have been calculated which gives the chances of risk involvement, chances of no risk involvement and chances of being doubtful (or further investigation), respectively, for the particular alternative. The average value for each region has been used to categorize the values of alternatives. Since POS(Ai) represents chances of risk involvement, the values less than average are more favorable. Thus we have rule F1 of Step 5. Similar criteria have 3. Generalized Hesitant Fuzzy Rough Sets 66 been adopted for negative and boundary regions according to their interpretations to get rules F2 and F3. Rules R1 to R3 have been developed to classify the alternatives into the best suitable region. Rule R2 has been developed due to the fact that as the distance between (favorable) values and their respective average values increases, the values become more favorable. At the end, alternatives are ranked according to the interpretation and preference of the three regions (POS, NEG and BND) by the decision maker. The above algorithm considers the original values given by the experts and classi es the alternatives based on the majority criteria, that is, those alternatives are selected which are better than average. Thus, three-way analysis using GHF RS provides comparatively the best ranking.

3.4.2 An illustrative example

One of the important real-world application area considered by many researchers of decision analysis is energy management. The goal is to evaluate and select the best energy project subject to the energy system performance [2, 26]. In this section we explain, with the help of an illustrative example, how the proposed method can be applied to select the best energy project. We consider the problem formulated by Xu and Xia [78] in which there are ve alternatives (energy projects) Ai (i = 1; 2; :::; n) to be invested and there are four risk factors involved: P1 : technological; P2 : environmental; P3 : sociopolitical; P4 : economic. There are several decision makers who evaluate the performance of the ve alternatives. The repeated values appear only once. Risk factors involved in the alternatives as per decision makers' evaluation have been given in the form of a hesitant fuzzy decision matrix in Table 3.3.

P1 P2 P3 P4 A1 .5,.4,.3 .9,.8,.7,.1 .5,.4,.2 .9,.6,.5,.3 f g f g f g f g A2 .5,.3 .9,.7,.6,.5,.2 .8,.6,.5,.1 .7,.4,.3 f g f g f g f g A3 .7,.6 .9,.6 .7,.5,.3 .6,.4 f g f g f g f g A4 .8,.7,.4,.3 .7,.4,.2 .8,.1 .9,.8,.6 f g f g f g f g A5 .9,.7,.6,.3,.1 .8,.7,.6,.4 .9,.8,.7 .9,.7,.6,.3 f g f g f g f g Table 3.3: Hesitant fuzzy sets representing risk factors

In real life problems, there are many constraints involved which cannot be avoided. These must be taken into consideration for reliability of the decision making process.

We assume that the projects A1 and A5 must be selected jointly, that is, if A1 is selected then A5 must be selected and vice versa. With the aid of decision making 3. Generalized Hesitant Fuzzy Rough Sets 67

POS NEG BND A1,A5 .5,.4,.3,.2,.1 .3,.2,.1 .89, .88, .86, .83, .80, .78, .75, f g f g .71,f .67, .63, .60, .57, .50, .40 g A2 .5,.4,.3,.2,.1 .1 .89, .88, .86, .83, .80 f g f g f g A3 .6,.5,.4,.3 .4,.3,.1 .86, .83, .80, .75, .57, .50, f g f g .43,f .40, .33, .25, .20, 0 g A4 .7,.6,.4,.3,.2,.1 .4,.3,.2,.1 .89, .88, .86, .83, .78, .75, .71, f g f g .67,f .63, .57, .56, .50, .43, .33, 0 g Table 3.4: Risk factors for POS, NEG and BND regions algorithm with GHF RS; we illustrate the selection procedure of energy projects as follows: Step 1: Input decision makers' evaluation in the form of hesitant fuzzy decision matrix presented in Table 3.3. These evaluations are in the form of hesitant fuzzy elements.

Step 2: Keeping in view the given constraint, that is, the projects A1 and A5 must be selected jointly, take set-valued map T : U P (U) as below: ! T (A1) = A1;A5 = T (A5);T (A2) = A2 ;T (A3) = A3 ;T (A4) = A4 : f g f g f g f g Step 3: For each alternative Ai (i = 1; 2; 3; 4; 5); compute the combined risk factor for positive, negative and boundary regions. (see Table 3.4). For example,

4 T (A1) = Z Z Pi (A0) H A T (A1) i=1 02 = P1(A1) P2(A1) P3(A1) P4(A1) f Z Z Z g P1(A5) P2(A5) P3(A5) P4(A5) Zf Z Z Z g = :5;:4;:3;:2;:1 :8;:7;:6;:4;:3;:1 f g Z f g = :5;:4;:3;:2;:1 = T (A5): f g H

Thus, POS(A1) = T (A1) = :5;:4;:3;:2;:1 = T (A5) = POS(A5): Also, H f g H 4 T (A1) = Y Y Pi (A0) H A T (A1) i=1 02 = P1(A1) P2(A1) P3(A1) P4(A1) f Y Y Y g P1(A5) P2(A5) P3(A5) P4(A5) Yf Y Y Y g = :9;:8;:7;:6;:5;:4;:3 :9;:8;:7 f g Y f g = :9;:8;:7 = T (A1): f g H c c Thus, NEG(A1) = (T (A1)) = :3;:2;:1 = (T (A5)) = NEG(A5): Also, BND(A1) = H f g H T (A1) T (A1) = :89;:88;:86;:83;:80;:78;:75;:71;:67;:63;:60;:57;:50;:40 : H H f g Similarly, the rest of entries can be calculated. 3. Generalized Hesitant Fuzzy Rough Sets 68

POS NEG BND A1,A5 .4333 .2800 .7173 A2 .4333 .1000 .8773 A3 .5600 .3733 .5667 A4 .5733 .3600 .6260 Average .499975 .278325 .696825

Table 3.5: Scores and average of scores for the expected risks

POS NEG BND Decision rule Corresponding values A1,A5 .4333 (F ) .2800 (F ) .7173 (F ) POS .4333 A2 .4333 (F ) .1000 (NF ) .8773 (F ) BND .8773 A3 .5600 (NF ) .3733 (F ) .5667 (NF ) NEG .3733 A4 .5733 (NF ) .3600 (F ) .6260 (NF ) NEG .3600

Table 3.6: classi cation of values as favorable or not favorable

Step 4: Calculate the scores of the expected risks for each alternative for the three regions in Table 3.4 as below: It was mentioned before De nition 1.3.6 that if HFEs (that are to be compared) have di erent lengths, the shorter one will be extended by adding maximum element to it as many times as it becomes of the same length. In Table 3.4, the HFE with maximum length contains fteen elements. Therefore, for comparison purpose, the other hesitant fuzzy elements will be extended up to fteen elements. For e.g., the HFE .5,.4,.3,.2,.1 will be extended to .5,.5,.5,.5,.5,.5,.5,.5,.5,.5,.5,.4,.3,.2,.1 and by f g f g De nition 1.3.6, its score will be calculated by adding these values and dividing by 15. Similarly, scores of rest of the HFEs will be calculated. Average of the values for each region thus obtained will be found using arithmetic mean. Results are shown in Table 3.5. Step 5: Based on these average values, each entry in Table 3.5 will be categorized as favorable (F ) or not favorable (NF ) according to the set of rules F1 to F3 and the alternatives will be classi ed into the suitable regions according to the decision rules

R1 to R3 as shown in Table 3.6. Step 6: Considering the corresponding values of the alternatives in Table 3.6, rank them according to the preference relation NEG BND P OS: Thus, the alternatives are ranked as: A3 A4 A2 A1 A5: The preference relation NEG BND POS has been chosen because value of a project in the negative region represents chances of no risk involvement, that is, chances of being good of that project, while in the positive region it represents chances of risk involvement, that is, chances for a project to be bad. The boundary region, on the other hand, is the non-commitment or doubtful region. Thus, the most suitable 3. Generalized Hesitant Fuzzy Rough Sets 69 preference order is NEG BND P OS: 3.4.3 Comparative analysis

Among others, the two most widely used decision analysis techniques are TOPSIS (technique for order preference by similarity to an ideal solution) [24] and AHP (analytic hierarchy process) [56]. The underlying concept in TOPSIS method is that the selected alternative should have the shortest distance from the positive ideal solution and farthest distance from the negative-ideal solution while AHP works by deriving the priorities for the performance of alternatives on each criterion based on pair-wise assessments of criteria. In below we brie y describe the various steps involved in TOPSIS and AHP .

TOPSIS

1. Construct a decision matrix for m-alternative and n-attributes.

2. Normalize the decision matrix with

xij rij = m 2 xij s i=1

for i = 1; 2; :::m and j = 1; 2:::n, where xij and rij are the original and normalized scores of decision matrix respectively.

3. Construct the weighted normalized decision matrix vij = wjrij, where wj is the weight for the j-attribute (criteria).

4. Determine the positive ideal solutions L = v ; v ; :::; v where v = max(vj) f 1 2 ng i f if j J and min(vj) if j J and negative ideal solutions L = v ; v ; :::; v 2 2 0g 0 f 10 20 n0 g where v = min(vj) if j J and max(vj) if j J . i0 f 2 2 0g 5. Find the separation measures for each alternative, 1=2 n + 2 S = (v vij) i 0 i 1 j=1 X ideal separation where i = 1; 2:::;@ m and A 1=2 n 2 S = (v0 vij) i 0 i 1 j=1 X negative ideal separation where @i = 1; 2; 3::::m. A 3. Generalized Hesitant Fuzzy Rough Sets 70

Numerical ratings Verbal judgments 1 Equally important (preferred) 3 Moderately more important 5 Strongly more important 7 Very strongly more important 9 Extremely more important

Table 3.7: Scales for prioritization process

6. Find the relative closeness to the ideal solution

Si Ci = + Si + Si

therefore 0 < Ci < 1. Select the alternative closest to 1.

AHP

1. Structure a hierarchy. De ne the problem, determine the criteria and identify the alternatives.

Overall Goal Select the best manufacturer

Delivery Criteria Cost Reliability time

Decision Company A Company B Company C alternatives

Fig. 3.2. Prototype of hierarchy structure in AHP

2. Make pair-wise comparisons. Rate the relative importance between each pair of decision alternatives and criteria. AHP uses 1 9 scale for the prioritization process. Intermediate numerical ratings of 2, 4, 6 and 8 can be assigned. If someone could not decide whether one criterion (alternative) is moderately more important then the other one or strongly more important than the other one, 4 (moderately to strongly more important) can be assigned. 3. Generalized Hesitant Fuzzy Rough Sets 71

3. Synthesize the results to determine the best alternative. Obtain the nal results. The output of AHP is the set of priorities of the alternatives.

Both of these techniques have been applied in many evaluation problems [2, 4], yet they are criticized for using crisp values in performance ratings and not re ecting the true human thinking style [25]. Linguistic variables can be more e ective in the problems that cannot be expressed accurately by quantitative values [48]. Thus, to overcome the said de ciencies, linguistic variables were used in fuzzy versions of these techniques [30, 70]. Besides their successful application in many domains [19, 31], yet their exibility to analyze real life situations is still doubtful as decision makers have to assign only one linguistic variable to each alternative subject to each criterion. Due to the exible nature of hesitant fuzzy sets, they can prove to be more worthwhile in the decision process as they allow more than one grading values to assess alternative with respect to a criterion [36, 37, 38, 89]. Two more approaches worth mentioning are three-way decision making by using decision theoretic rough sets (DTRS) (Yao et al. [85]) analyzed by Li et al. [32] and Liang et al. [33, 34]. The approaches provide reasonable semantic interpretation but use of equivalence relation in these techniques limit their practicality. The approach presented in this chapter is an e ort to develop a decision making technique with an aim of keeping essence of the existing techniques unaltered and also to cope with the situations where these techniques may not be applied. The salient advantages of the proposed approach over the existing ones can be listed as below:

1. The underlying concepts in TOPSIS and AHP have been retained by using the operators that are exible enough to maximize pro t (or minimize loss) and the alternatives are classi ed based on their relative importance.

2. By using hesitant fuzzy sets to construct decision matrix, the process becomes more exible and reliable and its three-way decision analysis nature reveals the true picture of the situation.

3. There are no cumbersome computations involved in it and it does not require equivalence classes as a prerequisite.

4. There are a lot of selection problems where more than one alternatives are to be selected keeping in view their mutual and joint performance. For example, if two persons are to be hired to work jointly on a project, their mutual understanding and ability to work jointly must be kept in view in order to run the project eciently. Secondly, in practical problems there may be certain constraints 3. Generalized Hesitant Fuzzy Rough Sets 72

Reference Technique or Measure Parameter or Weight Ranking Xu and Zhang [79] TOPSIS with hesitant w =(.17, .18, .35, .3) A3 A2 A4 A1 Xu and Zhang [79] TOPSIS with hesitant w =(.25, .25, .25, .25) A3 A4 A2 A1 Xu and Xia [78] hesitant Hausdor distance = 1 A3 A4 A2 A1 Xu and Xia [78] hesitant Hausdor distance = 10 A3 A4 A2 A1 Xu and Xia [78] hybrid hesitant distance = 1 A3 A4 A1 A2 Xu and Xia [78] hybrid hesitant distance = 10 A3 A2 A4 A1 Xu and Xia [78] hesitant ordered distance = 2 A3 A4 A2 A1 current chapter GHF RS NO A3 A4 A2 A1 Table 3.8: Comparison of ranking of alternatives

on the alternatives. For example, there may be limited subject combinations available for a student to opt. Existing techniques does not have a margin to handle such situations. This shortcoming has been overcome by introducing the set-valued map T to cluster the alternatives. The performance of alternatives subject to the given criteria will be assessed separately but to evaluate their combined performance and the expected output, they will be clustered using this map:

In the following, we compare the results of our illustrative Example 3.4.2 with two recently developed techniques of decision making with hesitant fuzzy sets proposed by Xu and Zhang [79] and Xu and Xia [78]. The former one is a generalization of TOPSIS in hesitant fuzzy environment while the later one is based on a variety of distance measures for hesitant fuzzy sets. In [79], the attribute weights are rst determined by establishing an optimiza- tion model based on the maximizing deviation method. Hesitant fuzzy positive and negative ideal solutions are then determined. In the next steps, separation of each alternative from ideal solutions and their relative closeness from positive ideal solution is calculated. On the basis of this relative closeness, the alternatives are ranked. Decision making in [78] is based on several distance measures that are used to calculate distance between the given alternatives and ideal alternative. Table 3.8 displays ranking of alternatives A1, A2, A3, A4 of Example 3.4.2 by the proposed and the mentioned techniques. The alternative A5 has been ignored because according to the constraint in our example, A1 and A5 are combined in one project. It can be seen that the ranking of alternatives by our technique is almost similar to the other techniques. It is worth mentioning that our methodology does not involve any parameter or weight vector and there are no heavy computations involved, while giving accurate results. 3. Generalized Hesitant Fuzzy Rough Sets 73

3.5 Conclusion

In this chapter, an approach is discussed to aggregate a given collection of hesitant fuzzy elements using lower and upper hesitant fuzzy approximation operators using a set-valued map T . Some basic properties of these approximations are studied. Three- way risk decision-making algorithm has been developed using GHF RS approach. This technique retains the properties of the renowned decision making techniques like TOPSIS, AHP and three way decision making using DTRS. In addition, it allows selection of clusters of alternatives (in case of compulsion) as well as individual ones. Instead of using an equivalence relation, this new model uses a set-valued map T to cluster elements of the universe which makes it more exible. A practical example of energy project selection is presented which demonstrates its signi cance. Compara- tive analysis depicts e ectiveness and precision of the methodology. A hesitant fuzzy topology is also introduced and hesitant fuzzy topological structure associated with these approximations is also investigated. The proposed approach may be useful for approximation of hesitant fuzzy elements when it is dicult to nd a hesitant fuzzy relation among the elements of a set under consideration. Chapter 4

Multi-granulation structure of fuzzi ed rough sets and GHFRS

4.1 Introduction

The rough sets proposed by Pawlak [46, 47] were based on approximating sets under lack of information using a single equivalence relation. The theory was further extended by using binary relations (serial, re exive or tolerance) instead of an equivalence relation to get more generalized forms of rough set theory. These theories were based on single granulation, that is, they used single binary relation to nd the required approximations. Later, Qian et al. [50] extended the approach presented by Pawlak by using more than one equivalence relations and name this as multi-granulation rough set theory. In this chapter we discuss multi-granulation structure of the generalized hesitant fuzzy rough set theory and fuzzi ed rough set theory presented in Chapters 3 and 4. The properties satis ed by the new approximation operators have been discussed and their relationship with the respective single granulation aproximation operators have also been explored.

4.2 Multi-granulation rough sets

The Pawlak's rough set theory is based on a single granulation. However, in situations where there is a contradiction or inconsistent relationship between it's values under one attribute set Q; Pawlak's approach is inapplicable. To deal with such situations, Qian et al. [50] presented the concept of multi-granulation rough sets.

74 4. Multi-granulation structure of fuzzi ed rough sets and GHFRS 75

De nition 4.2.1 [50] Let R1; R2; :::; Rn be n independent equivalence relations over a universe set U and A U: The lower and upper multi-granulation rough approximations of A in U are de ned respectively as

( Ri)(A) = x U [x]Ri A for some i = 1; 2; :::; n i=1 f 2 j g P and n

( Ri)(A) = x U [x]Ri A = for all i = 1; 2; :::; n : i=1 f 2 j \ 6 ; g P The boundary region of A U under MGRS environment is de ned as Bnd n (A) Ri i=1 n n P = ( Ri)(A) ( Ri)(A): i=1 n i=1 P P Some of the properties satis ed by MGR approximations are listed in the following theorem the proofs of which can be seen in [50].

Theorem 4.2.2 Let Ri i ; = 1; 2; :::; n where be a collection of equivalence re- f g 2I I f g lations over a universe set U. For any A U; the following properties hold: n n 1. ( Ri)(A) A ( Ri)(A); i=1 i=1 P P n n 2. ( Ri)( ) = = ( Ri)( ); i=1 ; ; i=1 ; P P n n 3. ( Ri)(U) = U = ( Ri)(U); i=1 i=1 P P n n c c 4. ( Ri)(A ) = ( Ri)(A) ; i=1 i=1 P P c n n c 5. ( Ri)(A ) = ( Ri)(A) ; i=1 i=1 ! P P n n n n n 6. ( Ri)(( Ri)(A)) = ( Ri)(( Ri)(A)) = ( Ri)(A); i=1 i=1 i=1 i=1 i=1 P P P P P n n n n n 7. ( Ri)(( Ri)(A)) = ( Ri)(( Ri)(A)) = ( Ri)(A); i=1 i=1 i=1 i=1 i=1 P P P P P n n 8. ( Ri)(A) = Ri(A); i=1 i=1 P [ 4. Multi-granulation structure of fuzzi ed rough sets and GHFRS 76

n n 9. ( Ri)(A) = Ri(A): i=1 i=1 P \ More properties can be seen in [50].

4.3 Multi-granulation fuzzi ed rough ( MGFR) sets In this section we extend multi-granulation rough set theory to multi-granulation fuzzi ed rough set theory. This extension is based on using a collection of tolerance fuzzy relations instead of a single tolerance fuzzy relation in fuzzi ed rough approximations.

De nition 4.3.1 Let Ri i be a collection of tolerance fuzzy relations over a uni- f g 2I verse set U characterized by their membership functions R : U U [0; 1]. For i ! (0; 1]; the multi-granulation fuzzi ed lower and upper rough approximations for a 2 given set X U are de ned as c Ri (X) = x U R (x; y) < for all y X i f 2 j i 2 g 2I i [2I and Ri (X) = x U R (x; y) for some y X i f 2 j i 2 g 2I i \2I

respectively. The pair (R (X); R (X)) is referred to as multi-granulation fuzzi ed rough set ( MGF RS). Positive, negative and boundary regions of X U for any (0; 1] are denoted and de ned as: 2

POS Ri (X) = Ri (X); i i 2I 2I c NEG Ri (X) = ( Ri (X)) ; i i 2I 2I BND Ri (X) = Ri (X) Ri (X): i i i 2I 2I 2I Based on the given information table, the elements in the positive, negative and boundary region can be interpreted, respectively as certain members, certain non- members and possible members (may or may not be) of X up to a relational degree of (0; 1]: Throughout, for MGF RS; the collection Ri i will be considered 2 f g 2I as tolerance fuzzy relation until otherwise speci ed.

4.3.1 Illustrative example

Let R1 and R2 be two fuzzy tolerance relations over a universe set U = x1; x2; x3; f x4; x5 as shown in Tables 4.1 and 4.2. g 4. Multi-granulation structure of fuzzi ed rough sets and GHFRS 77

R1 x1 x2 x3 x4 x5 x1 1 0.9 0.6 0.2 0.8 x2 0.9 1 0.7 0.3 0.5 x3 0.6 0.7 1 0.4 0.1 x4 0.2 0.3 0.4 1 0.8 x5 0.8 0.5 0.1 0.8 1

Table 4.1: Fuzzy tolerance relation R1

R2 x1 x2 x3 x4 x5 x1 1 0.8 0.2 0.4 0.7 x2 0.8 1 0.3 0.6 0.2 x3 0.2 0.3 1 0.5 0.9 x4 0.4 0.9 0.5 1 0.4 x5 0.7 0.2 0.9 0.4 1

Table 4.2: Fuzzy tolerance relation R2

The lower MGF R approximation of X = x1; x4; x5 with = 0:8 will be f g Ri (X) = x4; x5 as i f g 2I 0:8 c R1 (x4; y) < 0:8 for all y X ; 2 c and R (x5; y) < 0:8 for all y X : 1 2 Also, the upper MGF R approximation of X = x1; x4; x5 with = 0:8 will be f g Ri (X) = x1; x2; x4; x5 as i f g 2I 0:8

Ri (xj; xj) = 1 > 0:8 for all i = 1; 2 and j = 1; 4; 5,

R1 (x2; x1) = 0:9 > 0:8 and

R2 (x2; x1) = 0:8:

Thus, ( x4; x5 ; x1; x2; x4; x5 ) is 0:8 MGF RS. Also, f g f g

POS Ri (X) = x4; x5 ; i 2I f g NEG Ri (X) = x3 ; i 2I f g BND Ri (X) = x1; x2 : i 2I f g 4.3.2 Properties of MGF RS Proposition 4.3.2 Let Ri i be a collection of tolerance fuzzy relations over a uni- f g 2I verse set U and 1; 2 (0; 1] be such that 1 2: Then, 2 Ri (X) Ri (X) and i i 2I 1 2I 2 Ri (X) Ri (X): i i 2I 2 2I 1 4. Multi-granulation structure of fuzzi ed rough sets and GHFRS 78

Proof. For any x Ri (X); 2 i 2I 1 c R (x; y) < 1 for all y X for some i : i 2 2 I

Since 1 2; we have c R (x; y) < 1 2 for all y X for some i : i 2 2 I

Thus, x Ri (X) showing that Ri (X) Ri (X): 2 i i i 2I 2 2I 1 2I 2 Similarly, if x Ri (X); then 2 i 2I 2

R (x; y) 2 > 1 for some y X for all i : i 2 2 I

That is, x Ri (X) so that Ri (X) Ri (X): 2 i i i 2I 1 2I 2 2I 1

Theorem 4.3.3 Let Ri i be a collection of fuzzy tolerance relations over a universe f g 2I set U and (0; 1]: Then for X;Y U we have 2

(1) Ri (X) X Ri (X); i i 2I 2I

(2) Ri ( ) = = Ri ( ); i ; ; i ; 2I 2I

(3) Ri (U) = U = Ri (U); i i 2I 2I c c (4) Ri (X ) = ( Ri (X)) ; i i 2I 2I c c (5) Ri (X ) = ( Ri (X)) ; i i 2I 2I

(6) X Y = Ri (X) Ri (Y ) and Ri (X) Ri (Y ); ) i i i i 2I 2I 2I 2I

(7) Ri (X Y ) Ri (X) Ri (Y ); i \ i \ i 2I 2I 2I

(8) Ri (X Y ) Ri (X) Ri (Y ); i [ i [ i 2I 2I 2I

(9) Ri (X Y ) Ri (X) Ri (Y ); i [ i [ i 2I 2I 2I

(10) Ri (X Y ) Ri (X) Ri (Y ): i \ i \ i 2I 2I 2I Proof. (1) (3) are straightforward. 4. Multi-granulation structure of fuzzi ed rough sets and GHFRS 79

(4) For any x U; 2 c x Ri (X ) 2 i 2I c c R (x; y) < for all y (X ) for some i () i 2 2 I Ri (x; y) for any y X for some i () 6 2 c 2 I x = ( Ri (X)) x ( Ri (X)) : () 2 i () 2 i 2I 2I (5) For any x U; 2 c x Ri (X ) 2 i 2I c Ri (x; y) for some y X for all i () 2 c 2 I Ri (x; y) < for any y X for all i () 6 2 c 2 I x = Ri (X) x ( Ri (X)) : () 2 i () 2 i 2I 2I

(6) For any x Ri (X); 2 i 2I c R (x; y) < for all y X for some i : i 2 2 I In particular, since Y c Xc; we get c R (x; y) < for all y Y for some i ; i 2 2 I

which implies that x Ri (Y ): Thus, Ri (X) Ri (Y ): 2 i i i 2I 2I 2I Also, if x Ri (X); then 2 i 2I R (x; y) for some y X Y for all i : i 2 2 I

This implies that x Ri (Y ): Hence Ri (X) Ri (Y ): 2 i i i (7) Using part (6) above2I and the fact that2IX Y X;Y;2I we have \

Ri (X Y ) Ri (X); Ri (Y ) i \ i i 2I 2I 2I

and so, Ri (X Y ) Ri (X) Ri (Y ): i \ i \ i 2I 2I 2I (8) Since X;Y X Y; by part (6) of this theorem, we may write [

Ri (X); Ri (Y ) Ri (X Y ); i i i [ 2I 2I 2I and so, Ri (X) Ri (Y ) Ri (X Y ): i [ i i [ 2I 2I 2I (9) As X;Y X Y; using part (6) we get [

Ri (X); Ri (Y ) Ri (X Y ); i i i [ 2I 2I 2I Ri (X) Ri (Y ) Ri (X Y ): i [ i i [ 2I 2I 2I 4. Multi-granulation structure of fuzzi ed rough sets and GHFRS 80

(10) From part (6) and the fact that X Y X;Y; we have \

Ri (X Y ) Ri (X); Ri (Y ); i \ i i 2I 2I 2I and so, Ri (X Y ) Ri (X) Ri (Y ): i \ i \ i 2I 2I 2I

The following example shows that the inclusion in parts (7), (8), (9) and (10) in the above theorem are strict.

Example 4.3.4 Consider the tolerance fuzzy relations R1 and R2 over U = x1; x2; x3; x4; x5 f g as given in Tables 4.1 and 4.2 in Example 4.3.1 and let X = x1; x4; x5 and Y = f g x2; x4 be subsets of U. f g Taking = 0:8; we have

R1 + R2 (X Y ) = R1 + R2 ( x4 ) 0:8 \ 0:8 f g = fg R1 + R2 (X) = R1 + R2 ( x1; x4; x5 ) 0:8 0:8 f g = x4; x5 f g R1 + R2 (Y ) = R1 + R2 ( x2; x4 ) 0:8 0:8 f g = x4 : f g

This implies that R1 + R2 (X) R1 + R2 (Y ) = x4 * R1 + R2 (X Y ): 0:8 \ 0:8 f g 0:8 \ Taking = 0:8; we have

R1 + R2 (X Y ) = R1 + R2 ( x1; x2; x4; x5 ) 0:8 [ 0:8 f g = x1; x2; x4; x5 : f g But R1 + R2 (X) R1 + R2 (Y ) = x4; x5 x4 0:8 [ 0:8 f g [ f g = x4; x5 : f g This implies that R1 + R2 (X Y ) * R1 + R2 (X) R1 + R2 (Y ): 0:8 [ 0:8 [ 0:8 Again taking = 0:8; we have

R1 + R2 (X Y ) = R1 + R2 ( x4 ) 0:8 \ 0:8 f g = x4 f g R1 + R2 (X) = R1 + R2 ( x1; x4; x5 ) 0:8 0:8 f g = x1; x2; x4; x5 f g R1 + R2 (Y ) = R1 + R2 ( x2; x4 ) 0:8 0:8 f g = x1; x2; x4 : f g

This implies that R1 + R2 (X) R1 + R2 (Y ) = x1; x2; x4; x5 x1; x2; x4 = 0:8 \ 0:8 f g \ f g x2; x4 * R1 + R20:8(X Y ): f g \ 4. Multi-granulation structure of fuzzi ed rough sets and GHFRS 81

Now, taking = 0:7; we have

R1 + R2 (X Y ) = R1 + R2 ( x1; x2; x4; x5 ) 0:7 [ 0:7 f g = x1; x2; x3; x4; x5 f g R1 + R2 (X) = R1 + R2 ( x1; x4; x5 ) 0:7 0:7 f g = x1; x2; x4; x5 f g R1 + R2 (Y ) = R1 + R2 ( x2; x4 ) 0:7 0:7 f g = x1; x2; x4 : f g

Thus, R1 + R20:7(X) R1 + R20:7(Y ) = x1; x2; x4; x5 x1; x2; x4 = x1; x2; x4; x5 + [ f g[f g f g R1 + R2 (X Y ): 0:7 [

Proposition 4.3.5 Let Ri i be a collection of fuzzy tolerance relations over a uni- f g 2I verse set U and (0; 1]: Then for X U we have 2

1. Ri (X) = Ri (X); i 2I i [2I

2. Ri (X) = Ri (X): i 2I i \2I Proof. Straightforward.

Corollary 4.3.6 Let Ri i be a collection of fuzzy tolerance relations over a uni- f g 2I verse set U and (0; 1]: Then for X U we have 2

1. Rj (X) Ri (X) for any j ; i 2 I 2I

2. Ri (X) Rj (X) for any j : i 2 I 2I Proof. Follows from De nition 4.3.1 and Proposition 4.3.5.

Theorem 4.3.7 Let Ri i be a collection of fuzzy tolerance relations over a universe f g 2I set U and (0; 1]: Then for X;Y U we have 2

(1) Ri (X Y ) = Ri (X) Ri (Y ) ; i \ \ 2I i [2I (2) Ri (X Y ) = Ri (X) Ri (Y ) : i [ [ 2I i \2I Proof. (1) Using Proposition 4.3.5, we get

Ri (X Y ) = Ri (X Y ) i \ \ 2I i [2I = Ri (X) Ri (Y ) : \ i [2I 4. Multi-granulation structure of fuzzi ed rough sets and GHFRS 82

(2) Using Proposition 4.3.5, we get

Ri (X Y ) = Ri (X Y ) i [ [ 2I i \2I = Ri (X) Ri (Y ) : [

4.4 Multi-granulation singleton generalized hesitant fuzzy rough sets

In this section we'll discuss multi-granulation structure of singleton generalized hesitant fuzzy rough sets using a collection of set-valued mappings over a single universe set.

De nition 4.4.1 Let h be a hesitant fuzzy set on U; that is, h HF (U) and Ti : 2 f U P (U) i be a collection of set-valued mappings, where P (U) = P (U) . ! g 2I n; The lower and upper multi-granulation singleton approximations of h with respect to

Ti i are hesitant fuzzy sets Ti(h) and Ti(h) de ned for any x U as: f g 2I i i 2 2I 2I

Ti(h)(x) = Y Z h x0 i i 2I x0 Ti(x) ! 2I 2 and

Ti(h)(x) = Z Y h x0 i i 2I x0 Ti(x) ! 2I 2 respectively. The pair Ti(h); Ti(h) is called a multi-granulation singleton i i 2I 2I generalized hesitant fuzzy rough (MGSGHF R) set with respect to Ti i : h is f g 2I called de nable under this environment if Ti(h) = Ti(h). i i 2I 2I Example 4.4.2 Let U = x1; x2; x3; x4 be the universe set and h be a hesitant fuzzy f g set de ned over U as below:

h(x1) = 0:2; 0:7 ; h(x2) = 0:4; 0:6; 0:7 ; h(x3) = 0:3; 0:5; 0:9 ; h(x4) = 0:5; 0:8 : f g f g f g f g Let T1;T2 : U P (U) be two set-valued maps de ned over U as below: ! T1(x1) = x2; x3 ;T1(x2) = x1; x3 ;T1(x3) = x4 ;T1(x4) = x2; x4 ; f g f g f g f g T2(x1) = x1; x2 ;T2(x2) = x3; x4 ;T2(x3) = x1; x3 ;T2(x4) = x4 : f g f g f g f g The lower and upper MGSGHFR approximations of h with respect to T1;T2 are f g calculated below:

T1 + T2(h)(x1) = [h(x2) Z h(x3)] Y [h(x1) Z h(x2)] = :3;:4;:5;:6;:7 :2;:4;:6;:7 f g Y f g = :3;:4;:5;:6;:7 : f g 4. Multi-granulation structure of fuzzi ed rough sets and GHFRS 83

Similarly, the rest of the lower approximations can be calculated. The results are shown in below:

T1 + T2(h)(x2) = :3;:5;:7;:8 ;T1 + T2(h)(x3) = :5;:7;:8 ;T1 + T2(h)(x4) = f g f g :5;:6;:7;:8 : f g Also, T1 + T2(h)(x1) = [h(x2) Y h(x3)] Z [h(x1) Y h(x2)] = :4;:5;:6;:7;:9 :4;:6;:7 f g Y f g = :4;:5;:6;:7 : f g Similarly, T1 + T2(h)(x2) = :2;:3;:5;:7 ; T1 + T2(h)(x3) = :2;:3;:5;:7 ; T1 + T2(h)(x4) = f g f g :5;:6;:7;:8 : f g

Theorem 4.4.3 Let Ti : U P (U) i be a collection of set-valued maps over a f ! g 2I universe set U. For hesitant fuzzy sets h; h1 and h2; the following properties hold:

(1) Ti(h) h Ti(h) if Ti : U P (U) is re exive for all i ; i i ! 2 I 2I 2I

(2) Ti(h0) = h0 = Ti(h0); i i 2I 2I

(3) Ti(hU ) = hU = Ti(hU ); i i 2I 2I c c (4) Ti(h ) = ( Ti(h)) ; i i 2I 2I c c (5) Ti(h ) = ( Ti(h)) ; i i 2I 2I

(6) h1 h2 = Ti(h1) Ti(h2) and Ti(h1) Ti(h2); ) i i i i 2I 2I 2I 2I

(7) Ti(h1 h2) = Ti(h1) Ti(h2); i e i e i 2I 2I 2I

(8) Ti(h1 h2) = Ti(h1) Ti(h2); i d i d i 2I 2I 2I

(9) Ti(h1 h2) = Ti(h1) Ti(h2); i d i d i 2I 2I 2I

(10) Ti(h1 h2) = Ti(h1) Ti(h2): i e i e i 2I 2I 2I

Proof. (1) For any x U; since each Ti is re exive, using Remark 1.3.4, we have 2

Ti(h)(x) = h (x0) i Y Z i x Ti(x) 2I 2I 02 ! Y h(x) = h(x): i 2I 4. Multi-granulation structure of fuzzi ed rough sets and GHFRS 84

Also, h(x) Y h(x0) for all i : x Ti(x) 2 I 02 This implies

h(x) Z Y h x0 = Ti(h)(x): i i x0 Ti(x) ! 2I 2I 2 (2)

Ti(h0) = h0 (x0) i Y Z i x Ti(x) 2I 2I 02 !

= Y Z 0 = 0 = h0 i x Ti(x)f g f g 2I 02 !

= 0 = Ti(h0): Z Y i i x Ti(x)f g 2I 02 ! 2I (3)

Ti(hU ) = hU (x0) i Y Z i x Ti(x) 2I 2I 02 !

= Y Z 1 = 1 = hU i x Ti(x)f g f g 2I 02 !

= 1 = Ti(hU ): Z Y i i x Ti(x)f g 2I 02 ! 2I (4) For any x X; using part (5) of Theorem 1.3.5 we get 2

c c Ti(h )(x) = h (x0) i Y Z i x0 Ti(x) ! 2I 2I 2 c

= Y Y h (x0) i x0 Ti(x) ! 2I 2 c

= Z Y h (x0) i x0 Ti(x) !! 2I 2 c = Ti(h)(x) : i 2I (5) For any x U; using part (5) of Theorem 1.3.5 we get 2

c c Ti(h )(x) = h (x0) i Z Y i x0 Ti(x) ! 2I 2I 2 c

= Z Z h (x0) i x0 Ti(x) ! 2I 2 c

= Y Z h (x0) i x0 Ti(x) !! 2I 2 c = Ti(h)(x) : i 2I 4. Multi-granulation structure of fuzzi ed rough sets and GHFRS 85

(6) For any x U; using Corollary 3.2.7 we get 2

Ti(h1)(x) = h1 (x0) i Y Z i x Ti(x) 2I 2I 02 !

Y Z h2 (x0) i x Ti(x) 2I 02 ! = Ti(h2)(x): i 2I Also,

Ti(h1)(x) = h1 (x0) i Z Y i x Ti(x) 2I 2I 02 !

Z Y h2 (x0) i x Ti(x) 2I 02 ! = Ti(h2)(x): i 2I (7) For any x U; using Theorem 1.3.5 we get 2

Ti(h1 h2)(x) = (h1 h2)(x0) i e Y Z e i x Ti(x) 2I 2I 02 !

= Y Z (h1(x0) Z h2(x0)) i x Ti(x) 2I 02 !

= Y Z h1(x0) Z Z h2(x0) i x Ti(x) x Ti(x) 2I 02 ! 02 !!

= Y Z h1(x0) Z Y Z h2(x0) i x Ti(x) i x Ti(x) 2I 02 !! 2I 02 !! = Ti(h1)(x) Ti(h2)(x) i Z i 2I 2I = Ti(h1) Ti(h2) (x): i e i 2I 2I (8) For any x U; using Theorem 1.3.5 we get 2

Ti(h1 h2)(x) = (h1 h2)(x0) i d Y Z d i x Ti(x) 2I 2I 02 !

= Y Z (h1(x0) Y h2(x0)) i x Ti(x) 2I 02 !

= Y Z h1(x0) Y Z h2(x0) i x Ti(x) x Ti(x) 2I 02 ! 02 !!

= Y Z h1(x0) Y Y Z h2(x0) i x Ti(x) i x Ti(x) 2I 02 !! 2I 02 !! = Ti(h1)(x) Ti(h2)(x) i Y i 2I 2I = Ti(h1) Ti(h2) (x): i d i 2I 2I 4. Multi-granulation structure of fuzzi ed rough sets and GHFRS 86

(9) For any x U; using Theorem 1.3.5 we get 2

Ti(h1 h2)(x) = (h1 h2)(x0) i d Z Y d i x Ti(x) 2I 2I 02 !

= Z Y (h1(x0) Y h2(x0)) i x Ti(x) 2I 02 !

= Z Y h1(x0) Y Y h2(x0) i x Ti(x) x Ti(x) 2I 02 ! 02 !!

= Z Y h1(x0) Y Z Y h2(x0) i x Ti(x) i x Ti(x) 2I 02 !! 2I 02 !! = Ti(h1)(x) Ti(h2)(x) i Y i 2I 2I = Ti(h1) Ti(h2) (x): i d i 2I 2I (10) For any x U; using Theorem 1.3.5 we get 2

Ti(h1 h2)(x) = (h1 h2)(x0) i e Z Y e i x Ti(x) 2I 2I 02 !

= Z Y (h1(x0) Z h2(x0)) i x Ti(x) 2I 02 !

= Z Y h1(x0) Z Y h2(x0) i x Ti(x) x Ti(x) 2I 02 ! 02 !!

= Z Y h1(x0) Z Z Y h2(x0) i x Ti(x) i x Ti(x) 2I 02 !! 2I 02 !! = Ti(h1)(x) Ti(h2)(x) i Z i 2I 2I = Ti(h1) Ti(h2) (x): i e i 2I 2I

Proposition 4.4.4 For a hesitant fuzzy set h and a collection of set-valued mappings

Ti : U P (U) i over a universe set U, the following results hold. f ! g 2I

1. Ti(h) = (Ti(h)); i id 2I 2I

2. Ti(h) = (Ti(h)): i ie 2I 2I Proof. Straightforward.

Corollary 4.4.5 For a hesitant fuzzy set h and a collection of set-valued mappings

Ti : U P (U) i over a universe set U, we have f ! g 2I 4. Multi-granulation structure of fuzzi ed rough sets and GHFRS 87

1. Tj(h) Ti(h) for any j ; i 2 I 2I

2. Ti(h) Tj(h) for any j : i 2 I 2I Proof. Follows directly from Proposition 4.4.4 and Remark 1.3.4.

Proposition 4.4.6 For hesitant fuzzy sets h1; h2 and a collection of set-valued mappings Ti : U P (U) i over a universe set U, the following results hold: f ! g 2I

(1) Ti(h1 h2) = Ti(h1) Ti(h2) ; i e id e 2I 2I (2) Ti(h1 h2) = Ti(h1) Ti(h2) ; i d ie d 2I 2I (3) Ti(h1 h2) = Ti(h1) Ti(h2) ; i d id d 2I 2I (4) Ti(h1 h2) = Ti(h1) Ti(h2) : i e ie e 2I 2I Proof. Using Proposition 4.4.4 and Corollary 3.2.10, we have the following results: (1)

Ti(h1 h2) = Ti(h1 h2) i e id e 2I 2I = Ti(h1) Ti(h2) : id e 2I (2)

Ti(h1 h2) = Ti(h1 h2) i d ie d 2I 2I = Ti(h1) Ti(h2) : ie d 2I (3)

Ti(h1 h2) = Ti(h1 h2) i d id d 2I 2I = Ti(h1) Ti(h2) : id d 2I (4)

Ti(h1 h2) = Ti(h1 h2) : i e ie e 2I 2I = Ti(h1) Ti(h2) : ie e 2I

4.5 Conclusion

This chapter extends generalized hesitant fuzzy rough sets and fuzzi ed rough sets de ned in Chapters 3 and 4. The extension is based on de ning multi-granulation structure of both the techniques. In the former one, a collection of set-valued mappings 4. Multi-granulation structure of fuzzi ed rough sets and GHFRS 88 has been used, while in the later; a collection of fuzzy tolerance relations has been used to de ne the approximation operators. The properties of both the structures and relationship between single granulation and multi-granulation approximation operators have also been discussed. Chapter 5

Graphical equivalence relations and graphical partitions

5.1 Introduction

The notion of an equivalence relation on a set is an essential tool in many areas of Computer Science. In the theory of rough sets, equivalence relations model a type of indistinguishability. This leads to a theory in which a subset of a set equipped with an equivalence relation is approximated by a rough set which can be seen as a coarser or less detailed view of the original subset. Sets are, however, not the only structures which it is desirable to model at multiple levels of detail. In this chapter, we develop a theory of equivalence relations on graphs and hypergraphs motivated by the need to model these structures at di erent granularities. While granularity for graphs or networks is by no means a novelty, the approach of basing an approach on a generalization of equivalence relations is original. We use a notion of a relation on a graph, which we call graphical relations, for which we are able to de ne re exivity, transitivity and a weak form of symmetry. Properties of graphical relations satisfying these three properties are obtained and we establish their connection with graphical analogues of partitions and surjective functions.

5.2 Graphical Equivalence Relation and Graphical Par- tition

5.2.1 Graphs and Hypergraphs

We work with graphs which are undirected and which may have multiple edges between nodes as well as multiple loops on nodes. In a graph each edge is incident with

89 5. Graphical equivalence relations and graphical partitions 90 one or two nodes, but consideration of binary relations on graphs leads naturally to hypergraphs as we will see later. In a hypergraph there are edges and nodes, but each edge may be incident with any number of nodes. In our work we require that edges are incident with a non-zero number of nodes. One formalization of these structures is to have two disjoint sets for the nodes and for the edges. We use an alternative approach with a single set consisting of all the node and edges together with a relation which expresses the incidence between edges and nodes. When studying relations on these structures it is more convenient to use an equivalent de nition, in which there is a single set comprising both the edges and nodes together. This has been used in [63, 65] and is based on using a similar approach to graphs in [7].

De nition 5.2.1 A hypergraph consists of a set U and a re exive relation H U U (also called incidence relation) such that for all u; v; w U, if (u; v) H 2 2 and (v; w) H then u = v or v = w. 2 Given a hypergraph (U; H), an element u U is an edge if there is some v U 2 2 where (u; v) H and u = v. An element which is not an edge is a node. 2 6 It is straightforward to check that in a hypergraph (U; H) the relation H will be transitive as well as re exive so that it is a pre-order. Hypergraphs de ned in this way may have edges that are incident with non-empty sets of nodes and not just with one or two nodes as in the case of a graph. The decision not to allow edges incident with no nodes will be justi ed after we have de ned relations on hypergraphs. Graphs arise as a special case of hypergraphs as in the next de nition.

De nition 5.2.2 A graph is a hypergraph (U; H) which satis es the constraint that for every u U the set v U (u; v) H and u = v has at most two elements. 2 f 2 j 2 6 g A subgraph of a hypergraph (U; H) is de ned as a subset K U for which k K 2 and (k; u) H imply u K. 2 2 Three ways of visualizing hypergraph have been shown in Fig. (5.1-5.4). In the rst example we have a graph with a single edge, x, between two nodes x and y. This structure can be visualized as a poset or by drawing an edge between two nodes, or using a boundary to denote an edge which encloses the nodes with which it is incident. In the third example there are three nodes incident with one edge so that drawing an edge as a line segment with two ends is no longer possible. In the case of hypergraphs which are not graphs only the poset or boundary visualizations are e ective. 5. Graphical equivalence relations and graphical partitions 91

Poset Node-edge Boundary Hypergraph visualization visualization visualization x x y z

x U = x; y; z f g y z y z H = (x; x); (y; y); f (z; z); (x; y); (x; z) g Fig. 5.1. A hypergraph (graph) having one edge and two adjacent nodes

Poset Node-Edge Boundary Hypergraph visualization visualization visualization x w x w x w U = w; x; y; z y z f g y z H = (x; x); (y; y); z y f (z; z); (w; w); (x; y); (w; z) g Fig. 5.2. A hypergraph (graph) having two non-adjacent edges

Poset Node-Edge Boundary Hypergraph visualization visualization visualization w w U = w; x; y; z x y z f g H = (x; x); (y; y); x y z not possible f (z; z); (w; w); (w; x); (w; y); (w; z) g Fig. 5.3. A hypergraph with one edge and three adjacent nodes 5. Graphical equivalence relations and graphical partitions 92

Poset Node-Edge Boundary Hypergraph visualization visualization visualization v w w w x y v y U = v; w; x; y; z x f g z H = (x; x); (y; y); (z; z); v y z z x f (w; w); (v; v); (w; v); (w; y); (x; y); (x; z) g Fig. 5.4. A hypergraph (graph) with two adjacent edges

5.2.2 Relations on Hypergraphs

We follow the approach to relations on a hypergraph found in [62, 64]. We write the operation of composition of relations as ; and if R and S are binary relations on a set U we take R; S to mean composition in the following order

R; S = (u; w) U U v U((u; v) R and (v; w) S) f 2 j9 2 2 2 g De nition 5.2.3 Let (U; H) be a hypergraph, and R U U a relation on U. Then R is a Graphical Relation on (U; H) if H; R; H R. These relations are termed H-relations in [64], but the dependency on H can safely be left implicit in the present paper. The main properties of these operations are as follows, proofs of these can be found in [64]. We use R; S, R S, and R=S to denote n respectively the composition, the left residual , and the right residual of R and S as relations on U, that is, R S = R; S n R=S = R; S

where R denotes the converse of R and R denotes the complement of R.

Theorem 5.2.4 Let R and S be graphical relations on (U; H). Then the following properties hold.

1. R; S, R S, and R=S are all graphical relations. n 2. The identity relation 1 U U need not be a graphical relation, but H is a graphical relation and satis es H; R = R = R; H. 5. Graphical equivalence relations and graphical partitions 93

3. The converse R need not be a graphical relation, but there are graphical relations

a. R = H; R; H (the left converse of R), and

b. R = H R=H (the right converse of R). n The operations and form an adjoint pair.

4. The complement R need not be a graphical relation, but there are graphical relations

a. R = H; R; H (the pseudocomplement of R), and : b. R = H R=H (the dual pseudocomplement of R). n Fig. 5.5 shows relations on a graph with two nodes and a single edge. In the left hand example R is not a graphical relation and consists of two ordered pairs shown by arrows in the gure. The least graphical relation S containing R is shown on the right and consists of six ordered pairs. In general, if a graphical relation S relates a node n to an element x (either a node or an edge) then S must relate every edge incident with n to x. In addition if S relates an element x to an edge e then S must relate x to every node incident with e.

m j

Relation R which is The least graphical relation S not a graphical relation such that R S Fig. 5.5. Relations on a graph with two nodes and one edge

5.2.3 Graphical equivalence relations(GERs)

Our aim is to construct a generalization of equivalence relation that can serve as basis for de ning rough approximations of a graph.

De nition 5.2.5 A graphical relation R is said to be

1. re exive if H R, and 2. transitive if R; R R. 5. Graphical equivalence relations and graphical partitions 94

It is easily checked that for a graphical relation R the condition H R is equivalent to I R. Thus the graphical relations that are re exive and transitive in the above sense are just those relations on U which are re exive and transitive in the usual sense and which also meet the condition for being graphical relations. However this latter condition is unnecessary as the following lemma shows.

Lemma 5.2.6 Let R be any relation on U which satis es H R and which is transitive. Then R is a graphical relation.

Proof. We have H; R; H R; R; R R. The appropriate analogue of symmetry for graphical relations is not so easily obtained. As R need not be a graphical relation, we cannot demand that R = R for R to be symmetric. De ning the graphical analogue of an equivalence relation to be any equivalence relation which also a graphical relation is too strong as Theorem 5.2.9 below shows. In order to establish this result we need two lemmas.

Lemma 5.2.7 Let R be an equivalence relation on U and suppose R is also a graphical relation on (U; H). If the connected components of (U; H) are (Ci)i then for every 2I i; j either Ci Cj R or Ci Cj R = ?. 2 I \ Proof. As R is re exive H R, and as R is an equivalence relation the symmetric transitive closure of H, which we denote by H^ , is also contained within R. Now two elements u; v U lie in the same connected component of (U; H) if and only if 2 (u; v) H^ . Thus if v Ci and w Cj and (v; w) R, and in addition u Ci and 2 2 2 2 2 x Cj then we must have (u; x) R. 2 2 Lemma 5.2.8 Let S be an equivalence relation on . De ne a relation R on U as I follows. For any u Ci and v Cj put (u; v) R i (i; j) S. Then R is an 2 2 2 2 equivalence relation on U and a graphical relation.

Proof. It is straightforward to check that R is an equivalence relation since S is. To check that R is a graphical relation, suppose (u; v) R and that both (u ; u) and 2 0 (v; v ) lie in H. Since u and u belong to the same connected component, and similarly 0 0 for v and v , we must have that (u ; v ) R. 0 0 0 2 Theorem 5.2.9 There is a bijective correspondence between equivalence relations on the set of connected components of (U; H) and equivalence relations on U which are also graphical relations on (U; H).

Proof. We continue to use (Ci)i to denote the set of connected components of 2I (U; H). Suppose we have a graphical relation R which is also an equivalence relation on 5. Graphical equivalence relations and graphical partitions 95

U. De ne the relation R on by (i; j) R if and only if Ci Cj R. The relation I 2 R is re exive, since if u Ci then (u; u) R and so by Lemma 5.2.7, Ci Ci R. 2 2 It is routine work to check that R is both symmetric and transitive. By Lemma 5.2.8 the assignment R R is surjective as every equivalence relation on arises from 7! I some R. The assignment is injective by Lemma 5.2.7 since if R = S then both R and S can only be the relation Ci Cj. (i;j[) R 2 Theorem 5.2.9 shows that graphical relations on (U; H) which are also equivalence relations on U are too restrictive. In particular, in a graph with a single connected component the only such graphical relation is the universal relation U U. In the case of an ordinary relation R on U, symmetry is equivalent to R R. This, together with the fact that the left converse operation has already proved valuable in formulating a generalized notion of symmetry to establish the semantics of a modal logic based on graphical relations [66], motivates the next de nition.

De nition 5.2.10 A graphical relation R is said to be weakly symmetric if R R. In addition, R is said to be a graphical equivalence relation if it is re exive and transitive, in the sense of De nition 5.2.5, as well as weakly symmetric.

Graphical equivalence relations generalize the usual notion of equivalence relations on sets. We can regard a set U as the special case of (U; H) where H is the identity relation on U. In this case a graphical relation R on (U; H) is just a relation on U and R and R coincide so that:

Lemma 5.2.11 Suppose R is a relation on the set U. Then R is a graphical equivalence relation on (U; I) i R is an ordinary equivalence relation on U.

Proof. Straightforward. Further evidence that graphical equivalence relations provide a concept which plays the same role for graphs as the concept of equivalence relation does for sets is provided by the next observation.

Lemma 5.2.12 The identity relation H is a graphical equivalence relation.

Proof. H is a re exive and transitive. For the weakly symmetric condition, ^ ^ H H; H; H since H and hence H are re exive. 5. Graphical equivalence relations and graphical partitions 96

Fig. 5.6. All GERs on a graph with one edge and two incident nodes

To keep the things simple, all the arrows relating every element to itself and edge to its incident nodes have been omitted in gure 6, yet they should be assumed to be present.

5.2.4 Graphical Partitions(GPs)

In view of the well known correspondence between equivalence relations on U and partitions of U we next introduce the notion of graphical partitions on (U; H). In this section will always denote a set of non-empty subgraphs of the hypergraph (U; H) B and the elements of will be called blocks. B De nition 5.2.13 For each u U we de ne 2 u = K U u K and K : p q f j 2 2 Bg \ Note that if u lies in no element of then u is the intersection of an empty B p q collection of subsets of U. Thus in that case puq = U. In any case, puq must be a subgraph, and must contain u. The notation u does not make the role of the set p q B explicit, but this can always be determined from the context in the usage below.

Lemma 5.2.14 Let be any set of blocks, and let u; v U where v . Then B 2 p q 2 B u v if and only if u v . 2 p q p q p q Proof. We have u u , so u v implies u v : Conversely, suppose 2 p q p q p q 2 p q u v : We have v K U u K and K ; hence u v : 2 p q p q 2 f j 2 2 Bg p q p q Lemma 5.2.15 The following conditions on are equivalent. B 1. For every u U the subgraph u is an element of . 2 p q B 5. Graphical equivalence relations and graphical partitions 97

2. For every there is some such that = . C B D B C D Proof. To show the rst condition implies the second,T considerS the case of = . C 6 ; In this case we claim = u U u . Clearly u U u C fp q j 2 Cg C fp q j 2 Cg as u u . The reverse inclusion holds as u implies every block in contains 2 p q T S T2 C T S C T u so, as u is the smallest block in containing u, we must have u . In p q B T p q C the case that = so that = U, the rst condition guarantees that = U as C ; C BT u u always holds, so we can take = . 2 p q T D B S Assume now the second condition and consider = K u K . The inter- C f 2 Bj 2 g section = u is expressible as a union of elements of at least one of which, B C p q B say, contains u. Now B as B , and B as B belongs to a set . T C 2 C C D B u B Hence we have p q = T. T De nition 5.2.16 The set is said to be a graphical partition if the following B conditions are satis ed.

P1: For every u U the subgraph u is an element of . 2 p q B P2: Let B and . If B = then B . 2 B C B C 2 C P3: For any u u ; there exist v;S v U such that 2 p 0q 0 2

u0Hv0 and vHu and v0 v : 2 p q We make some observations on these conditions. The condition P1 does not require that the blocks are closed under intersections in general, only that intersecting all the blocks containing a given element yields a block. The condition P2 expresses the requirement that no block is the union of strictly smaller blocks. To explain P3, in the case of an ordinary equivalence relation on U, and denoting the equivalence class of u by [u], we have u [u ] implies u [u]. Condition P3 generalizes this property 2 0 0 2 by replacing the equivalence classes [u] and [u0] by puq and pu0q and weakening the symmetric interchange of u and u0.

Lemma 5.2.17 Let be any collection of blocks which satis es P1. Then satis es B B P2 if and only if every block is of the form puq for some u.

Proof. Assume P2. Let B be any block and let u B. Since B K U 2 B 2 2 f j u K and K we have puq B. Thus we can write B = u Bpuq. Hence by 2 2 Bg 2 P2 B u u , must have the form p q for some . S Conversely, if puq = v V puq for some V U then for all v V we have 2 2 v u . But as u u there must be some particular v V for which u v p q p q 2 p Sq 2 2 p q which implies u v so that u = v . p q p q p q p q 5. Graphical equivalence relations and graphical partitions 98

Observe that a simple example where P1 and P3 are satis ed but not P2 is U = 1; 2 , H is the identity relation on U, and = 1 ; 2 ; 1; 2 . Sets of blocks f g B ff g f g f gg satisfying only P1 and P2 are thus not necessarily partitions of the set of nodes of a graph which has no edges. However graphical relations do have this property which is important if they are to be considered as the result of generalizing equivalence relations from the context of sets to that of graphs and hypergraphs.

Theorem 5.2.18 Suppose is a graphical partition of (U; H) and that H is the iden- B tity relation on U. Then is a partition of the set U. B Proof. Every u U lies in some block since u u . Suppose u v = 2 2 p q p q \ p q 6 ; and that u lies in this intersection. Then u u so by P3 u u . By Lemma 0 0 2 p q 2 p 0q 5.2.14, these two facts imply respectively that u u and u u . Hence p 0q p q p q p 0q every element of U lies in exactly one block and we have a partition. Fig. 5.7 shows all graphical partitions of graphs consisting of a single node and two nodes, while all graphical partitions of a graph with one edge and two attached nodes are shown in g. 5.8. An alternative visualization of graphical partitions can be seen in g. 5.9.

Fig. 5.7. All graphical partitions of graphs with single node and two nodes only

Fig. 5.8. All graphical partitions of a graph with one edge and two nodes 5. Graphical equivalence relations and graphical partitions 99

Fig. 5.9. Alternative visualization of all graphical partitions of a graph with one edge and two nodes obtained by drawing the edge as a boundary enclosing its incident nodes

5.2.5 Correspondence between graphical equivalence relations and graphical partitions

In the set theoretic context, every partition induces an equivalence relation and every equivalence relation induces a partition. Seeing how the concepts of graphical equivalence relation and graphical partition are related to each other is our next task.

We start by de ning a relation from a partition by relating u0 to u when every block containing u0 contains u.

Theorem 5.2.19 For a given graphical partition of a graph G, the relation R de- B ned as follows is a graphical equivalence relation:

u0Ru if and only if u0 u : p q p q Proof. First check that R is graphical. Suppose that xHy, yRz, and zHw. We need to show x w . Since every block containing x also contains y we have p q p q x y and similarly z w so we get x w since y z by de nition p q p q p q p q p q p q p q p q of R. Transitivity is immediate since has these properties. By Lemma 5.2.14 the condition u u is equivalent to u u . Thus to verify weak symmetry we p 0q p q 2 p 0q need to show that if u u then there exist v; v where u Hv , and v v and 2 p 0q 0 0 0 0 2 p q vHu. This is assured by P3. For re exivity we need H R. If uHv then, since every 5. Graphical equivalence relations and graphical partitions 100 block is a subgraph, every block containing u must also contain v. Thus v u so 2 p q that v u and hence uRv. By Lemma 5.2.6, R is a graphical relation. p q p q The above result shows that given any partition on a graph G, we can construct a graphical equivalence relation induced by that partition. We now consider the reverse process.

Theorem 5.2.20 Let R be a graphical equivalence relation on (U; H). Then the family of all subsets of the form Ru = v U uRv is a graphical partition of (U; H). f 2 j g Proof. Every set of the form Ru is a non-empty subgraph since R is a re exive graphical relation. First we show that u = Ru. By de nition u = Rv U p q p q f j u Rv . As u Ru by re exivity, we have u Ru. But if w Ru and u Rv for 2 g 2 p q 2 T 2 some v and w then vRu and uRw so that w Rv by transitivity. Thus every w Ru 2 2 lies in every Rv containing u so Ru u . Hence u = Ru, which establishes P1. p q p q For P2, let V U and assume Ru = Rv U v V . We must have f j 2 g u Rv = v for some v V . By the assumption v u , but u v by 2 p q 2 S p q p q p q p q Lemma 5.2.14. For P3, suppose that u u . This means u Ru so that by weak symmetry 2 p 0q 0 there exist v; v U such that u Hv and v Rv and vHu. But from vRv we get 0 2 0 0 0 0 v Rv = v which establishes P3. 0 2 p q

5.3 Graphical Partitions as Surjective Mappings

Given a partition of a set U where is the set of equivalence classes, it is well known B that we have a surjective function ' : U where ' takes each u U to its equiv- !B 2 alence class. We will call this function the classi cation function for the partition. If : U V is any surjective function, then we obtain an equivalence relation R on ! U by de ning uRu0 if and only if u = u0. In the case that V is the set of blocks of a partition, , and that is the associated classi cation function, then the relation R B will be the equivalence relation for the partition . Because of these facts, it it often B useful to regard partitions of a set U as essentially equivalent to surjective functions with domain U, although in some circumstances it may be important to be aware that the same partition can be associated to distinct surjective functions. In this section we show how this situation generalizes to graphs and hypergraphs.

De nition 5.3.1 Let (U; H) and (U ;H ) be preorders and suppose that ' :(U; H) 0 0 ! (U 0;H0) is surjective and order preserving. Then we say ' is a graphical classi cation if for all u; v U such that 'uH 'v there exist u+; v U where 2 0 2 1. uHu+, 5. Graphical equivalence relations and graphical partitions 101

2. vHv, and

+ 3. 'vH0'u .

Theorem 5.3.2 Let be a graphical partition on (U; H) then the function ' :(U; H) B ! ( ; ) where u u is a graphical classi cation. B 7! p q Proof. To show is order preserving, let u; v U and uHv. Since u is a 2 p q subgraph, we have v u so u v by Lemma 5.2.14. The remaining condition is a 2 p q straightforward consequence of P3. Given a graphical classi cation we can obtain a graphical partition as follows.

Theorem 5.3.3 Suppose ' :(U; H) (U ;H ) is a graphical classi cation. For each ! 0 0 u U de ne u' = v U 'uH 'v . Then u' U u U is a graphical partition 2 f 2 j 0 g f j 2 g of (U; H).

Proof. Each block u is non-empty as it contains u by the re exivity of H0. To check that u is a subgraph, suppose v u and vHw. From v u we get uH v 2 2 0 and as is order preserving we get vH w from vHw. Hence uH w so that w u. 0 0 2 Next we show that u = u . As u u we have v u . This gives p q u v 2 2 u u. For the reverse inclusion, let w u and v be such that u v. These p q 2 T 2 imply that w v since vH u and uH w. 2 0 0 From u = puq we not only get P1, but as all the blocks are of the form puq, so we have P2 by Lemma 5.2.17. Finally P3 follows by a routine calculation from the de nition of graphical classi cation. 5. Graphical equivalence relations and graphical partitions 102

Fig. 5.10. All graphical classi cations for a graph with one edge and two nodes

5.4 Two-tier Graphical Partitions

The equivalence classes of a set U are subsets of U and together they constitute another set. In the case of graphs, the blocks of a graphical partition are subgraphs and it would be natural to expect that collectively they form another graph. By looking at the graphical classi cations in g 5.10 we can see that the blocks with ordered by inclusion do form a partial order but not always one that can be regarded as a graph.

The problem cases are those where we have blocks K1;K2;K3 where K1 ( K2 ( K3. De nition 5.4.1 Suppose is a graphical partition of (U; H). We say is a two- B B tier graphical partition of (U; H) when for all K1;K2;K3 , 2 B if K1 K2 K3 then K1 = K2 or K2 = K3: (TTP) To state the equivalent form of this condition for a graphical equivalence relation R, we recall from [64] that graphical relations are closed under the converse complement operation aR = R = R. De nition 5.4.2 Suppose R is a graphical equivalence relation on (U; H). We say R is a two-tier graphical equivalence relation on (U; H) when

(R R); (R R) = ?: (TTR) \ a \ a 5. Graphical equivalence relations and graphical partitions 103

For an ordinary relation R on set U symmetry is equivalent to R R = ?. Thus \ a in the case that H is the identity relation on U every graphical equivalence relation is two-tier and following Lemma 5.2.11 we have the next result.

Lemma 5.4.3 Suppose R is a relation on the set U. Then R is a two-tier graphical equivalence relation on (U; I) if and only if R is an ordinary equivalence relation on U.

Proof. Straightforward. For graphical relations the condition R R = ? clearly implies weak symmetry, \ a but the condition TTR is independent of weak symmetry.

Theorem 5.4.4 There is a bijective correspondence between two tier graphical partitions and two tier graphical equivalence relations. In particular:

1. Given a two-tier graphical partition of (U; H), the relation R( ) de ned as B B follows is a two-tier graphical equivalence relation:

u0R( )u if and only if u0 u : B p q p q

2. Let R be a two-tier graphical equivalence relation on (U; H). Then the family of all subsets of the form Ru = v U uRv is a two-tier graphical partition of f 2 j g (U; H) which we will denote by |(R).

3. For any two tier graphical partition and any two tier graphical equivalence B relation R, both R(|(R)) = R and |(R( ) = : B B Proof.

1. By Theorem 5.2.19 we only need to show that R( ) is two-tier. Let xR( )y and B B yR( )z. Thus pxq pyq pzq so either pxq pyq or pyq pzq and hence B ` ` either yR( )x or zR( )y. Thus at least one of xR( )y and yR( )z does not B B B B hold.

2. By Theorem 5.2.20 we only need to show that B(R) is two-tier. Suppose that Ru Rv Rw. Then we have uRv and vRw. Since (R R ); (R R ) = , at \ \ ; least one of vRu and wRv must hold. Hence either Ru = Rv or Rv = Rw.

3. We use u to denote the least block in containing u. The blocks in B(R( )) p q B B are the sets of the form v U uR( )v for u U. But since uR( )v if and f 2 j B g 2 B only if v u if and only if v u , the sets v U uR( )v are exactly p q p q 2 p q f 2 j B g 5. Graphical equivalence relations and graphical partitions 104

the blocks of . To show R(B(R)) = R we have that for any u; v U, uR(|(R))v B 2 if and only if u v where blocks are with respect to R(B(R)). But this p q p q holds if and only if Ru Rv if and only if for all w U; vRw uRw: (5.1) 2 ) From (5.1) we get uRv since vRv. Conversely, if uRv then (5.1) holds by transitivity.

Theorem 5.4.5 Let be a set of non-empty subgraphs of (U; H). Then is a two- B B tier graphical partition if and only if it satis es P1, TTP, and for all K1;K2 2 B where K1 K2 and K1 = K2, at least one of the following holds. 6

1. K2 K1 contains only edges

2. Every node in K1 is incident with some edge in K2 and K1 consists only of nodes.

Proof. First suppose that is a two-tier graphical partition. We need to show B that one of the two conditions in the statement of the theorem holds. Suppose we have blocks K1, K2 where K1 is a proper subset of K2 and n is a node in K2 K1. We must have K2 = n since n K2 follows from n K2 and the reverse p q p q 2 inclusion holds as is two-tier. Now the subgraph K1 must contain nodes, so let B m K1 be any node. We must have K1 = m so from m n we know by P3 2 p q p q p q that there are n ; m where nHn , and m Hm and n m . Since n is a node 0 0 0 0 p 0q p 0q nHn implies n = n. So from n m we must have m K2 K1, and since 0 0 p 0q p 0q 0 2 m Hm we know m is an edge as the only node it could be is m K1. Thus m is 0 0 2 incident with an edge in K2 and we need next to show that K1 consists only of nodes.

Suppose that e is an edge in K1. As e n we have by P3 the existence of e and p q p q 0 n0 where nHn0 and e0He. As n is a node and e is an edge these imply n = n0 and e = e . But as P3 also gives n e , we have a contradiction as n e implies 0 p 0q p 0q p q p q e K2. 2 For the converse we need to show that any set of blocks satisfying at least one B of (1) and (2) as well as P1 and TTP is a two-tier graphical partition. Suppose that has these properties. We need to show that it satis es P2 and P3. By Lemma B 5.2.17 we can show that every block has the form puq to justify P2. So suppose we have a block B and that u B. If u ( B then by conditions (1) and (2), either 2 p q B u contains only edges, which must necessarily be incident with nodes in u , or p q p q u contains only nodes which are all incident with edges in B u . Thus in either p q p q 5. Graphical equivalence relations and graphical partitions 105 case, there is some edge e where u ( e B, so we must have B = e by TTP. p q p q p q To justify P3, suppose we have u; v U where u v . We need to nd u and 2 p q p q 0 v satisfying vHv and u Hu and v u . If u = v we can take u = u and 0 0 0 p 0q p 0q p q p q 0 v = v, so assume that u = v . There are four cases according as each of u and v 0 p q 6 p q is an edge or a node. u node; v node. Put v0 = v and u0 any edge in pvq incident with u. u node; v edge. If v u consists entirely of edges, then take v to be a node in p q p q 0 puq incident with v and take u0 to be u. If puq consists entirely of nodes incident with edges in v u , then take u to be any edge incident with u and take v p q p q 0 0 to be v. u edge; v node. This cannot arise since v being a node means that v u cannot p qp q consist entirely of edges. But then puq has to consist entirely of nodes. u edge; v edge. Take u0 to be u and v0 to be any node in puq incident with v. 5. Graphical equivalence relations and graphical partitions 106

g g

g g g

g g g g

g g

Fig. 5.11. All isomorphism classes of two-tier graphical equivalence relations on a graph with two edges and three nodes 5. Graphical equivalence relations and graphical partitions 107

5.5 Conclusion

Our account started with a notion of graphical equivalence relation which can serve as a basis to de ne rough approximations of graphs and hypergraphs. The notion of graphical partitions is then de ned and the correspondence between graphical equivalence relations (GERs) and the graphical partitions (GP s) is then established. Surjec- tive functions equivalent to graphical partitions have also been de ned and discussed. Also, the collection of partition blocks of a hypergraph that again form a graph or a hypergraph are exclusively discussed. We have shown how to generalize the correspondence between partitions and equivalence relations on sets to the context of graphs and hypergraphs. One direction for further work is to use the foundations established here to build a theory of rough graphs which generalizes that of rough sets. In the same way that symmetry for relations on graphs is much more subtle than it is for sets, it seems likely that the basic constructions in rough set theory will have multiple generalizations to the graph case. There are likely to be many di erent ways to proceed and the choice between these will need to be guided by the need of applications and not just by theoretical elegance. The correspondence between relations on a set and certain functions on the power set is also important in mathematical morphology [6, 59] which has established connec- tions with rough set theory [5, 63]. Mathematical morphology provides an approach to images at di erent levels of detail where the level of detail is parameterized, in the simple case of black and white images, by a relation on the set of all pixels. Re- cent developments in mathematical morphology study situations in which the pixels in an image form the nodes of a graph as opposed to just being a set [14, 42]. It thus seems likely that the results in this chapter will have relevance to understanding mathematical morphology on graphs. Chapter 6

Rough graphs and their algebraic structure

6.1 Introduction

Extensive research has been done on de ning lower and upper rough approximations of a set by using one or more relations using various techniques. But the study of approximating structures like graphs instead of sets needs further investigation. In this chapter, we develop lower and upper rough approximations of a graph (which is a particular type of pre-order) by using graphical equivalence relation (or equivalently graphical partition) discussed in the previous chapter. Furthermore, we de ne the notion of rough graphs and explore their structural and algebraic properties.

6.2 Rough graphs

De nition 6.2.1 Given any subgraph K of a graph (G; H) (or simply G), its upper and lower approximations with respect to a graphical equivalence relation R are de ned as follows. The notation u means v G uRv and we'll call it the partition block p q f 2 j g of G generated by u.

K = u G u K Lower Approximation fp q jp q g K = u G u K Upper Approximation S fp q j 2 g This notation doesS not explicitly reference the relation R, however this will not cause diculties as R can always be inferred from the context in our usage below. If more than one relation is involved in the discussion subscripts can be introduced to disambiguate as necessary.

108 6. Rough graphs and their algebraic structure 109

De nition 6.2.2 For any subgraph K, the pair of its lower and upper approximations, (K; K) is called the rough graph for K with respect to R. The subgraph K is called de nable if K = K = K.

Note that following usual case of rough sets we might expect to be able to write

K = u G u K = ; f 2 jp q \ 6 ;g K = u G u K ; f 2 jp q g However the rst of these is not equivalent to the expression in De nition 6.2.1 as the example in g. 6.1 shows.

m n e

Fig. 6.1. Example where n m p q \ f g = but n = m : 6 ; p q 6 p q Here the whole graph has nodes m and n and an edge e. Taking the subgraph K = m f g we nd n K = but according to De nition 6.2.1 n = K. Nevertheless we do have p q \ 6 ; 2 the following.

Example 6.2.3 Consider a graph G with graphical equivalence relation R as shown in g. 6.2. To avoid complexity, the arrows relating every element to itself and every edge to its incident nodes have been omitted but they should be assumed to be present in all the gures in this chapter unless otherwise stated.

e1 e3 n1 n2 e2 n3 n4 e4 n5

Fig. 6.2. Graph G with graphical equivalence relation R

The eight partition blocks of G with respect to R, that is, u = v G uRv for p q f 2 j g all u G; are shown in g. 6.3. 2 6. Rough graphs and their algebraic structure 110

n1 n2 n3 n5 n1 e1 n2

n2 e2 n3 n4 n2 e2 n3 e3 n4

n2 e2 n3 n4 e4 n5

Fig. 6.3. All partition blocks of graph G under R

Let L be a subgraph as shown in Fig 6.4. The same gure shows its lower and upper approximations according to De nition 6.2.1.

n3 e3 n4 e4 n5 L

n3 n5 L

n2 e2 n3 e3 n4 e4 n5 L Fig. 6.4. Subgraph L of G with its upper and lower approximations

The two approximations have been calculated as below: By De nition 6.2.1, lower approximation of L will be the union of all those blocks that are contained in L. By looking at Fig 6.3, we can easily see that only the blocks

n3 and n5 are contained in L: Thus L = n3; n5 : By the same de nition, upper f g f g f g approximation of L will be the union of all the blocks that are generated by the elements of L. Again from Fig. 6.3, the blocks generated by the elements of L are: pn3q = n3 ; n5 = n5 ; n4 = n2; e2; n3; n4 ; e3 = n2; e2; n3; e3; n4 and e4 = f g p q f g p q f g p q f g p q n2; e2; n3; n4; e4; n5 . Their union gives the upper approximation of L, that is, L = f g n2; e2; n3; e3; n4; e4; n5 : f g Lemma 6.2.4 For any subgraph, K, u G u K = u G u K : f 2 jp q g [fp q jp q g 6. Rough graphs and their algebraic structure 111

Proof. If u K and x u then x u so u G u K u p q 2 p q p q p q fp q jp q g f 2 G u K : The reverse inclusion holds as u u . jp q g 2 p q [ Lemma 6.2.5 Let V be any subset of G. Then

puq = puq = puq: u V u V u V [2 [2 [2 Proof. For the rst equation we have this calculation.

u = v G v u p q fp q jp q p qg u V u V [2 [ [2 = v G v V fp q j 2 g = [ puq u V [2 For the second, we have by de nition that

u = v G v u : p q fp q j 2 p qg u V u V [2 [2 Now if w v for some v u ; then w v for some v u where u V . 2 p q 2 p q 2 p q 2 p q 2 u V [2 But v u implies v u , so w u for some u V . Hence w u 2 p q p q p q 2 p q 2 2 fp q G u V = u : [ j 2 g p q u V We now list[2 properties of these graphical approximations.

Theorem 6.2.6 For a graphical equivalence relation R over a graph G, the following properties hold for any subgraphs K, L of G.

1. K K K; 2. If K L then K L and K L; 3. = = , G = G = G;

4. K L = K L; \ \ 5. K L = K L; [ [ 6. K L K L; [ [ 7. K L K L; \ \ 8. (K) = K; 6. Rough graphs and their algebraic structure 112

9. K = K;

10. (K) = K;

11. K = K. Proof. (1) Follows from the fact that u K implies u K. p q 2 (2) (7) are straightforward from the de nitions. (8) (11) are all consequences of Lemma 6.2.5 since K and K are both by de nition unions of blocks.

Remark 6.2.7 In classical rough set approximations, properties 10 and 11 in the above theorem hold if R is symmetric. But here the condition of weakly symmetric on R is sucient for the properties to hold.

The following example shows that the reverse inclusions in parts 6 and 7 in Theorem 6.2.6 may not hold.

Example 6.2.8 Consider the graph G with graphical equivalence relation R as shown in Fig. 6.2 in Example 6.2.3. Let K and L be subgraphs of G as shown in Fig. 6.5. Their upper and lower approximations have been shown in the same gure.

n2 e2 n3 n5 n3 e3 n4 K L

n2 n3 n5 K L n3

n2 e2 n3 n4 n5 n2 e2 n3 e3 n4 K L Fig. 6.5. Two Subgraphs of G with their approximations

n2 e2 n3 e3 n4 n5 K L [

n2 e2 n3 e3 n4 n5 n2 n3 n5 K L K L [ [ Fig. 6.6. Lower approximation of union of subgraphs and union of their lower approximations 6. Rough graphs and their algebraic structure 113

K L n3 \

n2 e2 n3 n4 K L n3 K L \ \ Fig. 6.7. Upper approximation of intersection of subgraphs and intersection of their upper approximations

Figures 6.6 and 6.7 show that K L = K L and K L = K L: Thus, the [ 6 [ \ 6 \ inclusions in parts 6 and 7 in Theorem 6.2.6 are strict.

6.3 Algebraic structure of rough graphs

Many algebraic structures have been de ned in connection with rough approximations [15, 22, 49, 60, 73]. Some of these algebras satisfy the law of excluded middle while others violate it. Since, in the case of graphical approximations, law of excluded middle is not satis ed due to the unavailability of (set theory) complement, we'll discuss here two algebras namely Stone algebra and Heyting algebra in which law of excluded middle does not hold. We'll try to explore their connection with rough graph approximations. Pomykala and Pomykala [49] explored Stone algebra associated with rough sets. They proved that the family of rough sets form a complete and atomic Stone algebra. Pagliani [45] discussed Heyting algebras in connection with rough sets. We rst recall the de nitions of lattice, Stone and Heyting algebras.

De nition 6.3.1 A semilattice (A; ; 1) is a commutative monoid where x x = x for all x A. 2 De nition 6.3.2 A lattice is an algebra (A; ; ; ; ) such that _ ^ ? | 1. both (A; ; ) and (A; ; ) are semilattices, and _ ? ^ | 2. for all x; y A, x y = y if and only if y x = x: 2 _ ^ In a lattice we write for the partial order de ned by x y if x y = y. _ De nition 6.3.3 A distributive lattice is a lattice A satisfying the identity 6. Rough graphs and their algebraic structure 114

x (y z) = (x y) (x z) for all x; y; z A: ^ _ ^ _ ^ 2 De nition 6.3.4 An algebra (S; ; ; ; 0; 1) with two binary operations and , ^ _ : _ ^ a unary operation , and two nullary operations 0 and 1 is a Stone algebra if it : satis es, for any x S: 2 S1: (S; ; ) is a distributive lattice ^ _ S2: x 0 = 0 and x 1 = 1 ^ _ S3: x x = 0 ^ : S4: x y = 0 implies y x ^ : S5: x x = 1: : _ :: De nition 6.3.5 An algebra (S; ; ; ; ; 0; 1) is a double Stone algebra if (S; ; ; ; 0; 1) ^ _ : ^ _ : is a Stone algebra and, additionally, is a unary operation that satis es for any x S: 2 S6: x x = 1 _ S7: x y = 1 implies x y _ S8: x x = 0 ^ Operation is called a dual pseudocomplementation. De nition 6.3.6 A Heyting algebra is an algebra (A; ; ; ; ; ) such that _ ^ ! ? > H1: (A; ; ; ; ) is a lattice, _ ^ ? > H2: for all x; y; z A; x y z if and only if x y z. 2 ! ^ Lemma 6.3.7 An algebraic structure (A; ; ; ; ; ) is a Heyting algebra if and _ ^ ! ? > only if x y is the greatest element a, such that x a y for all x; y H. ! ^ 2 De nition 6.3.8 An algebra (H; ; ; ; ; 0; 1) is a double Heyting algebra, if ^ _ ! (H; ; ; ; 0; 1) is a Heyting algebra and additionally is a binary operation that ^ _ ! satis es, for any x; y; z H: 2 H3: x x = 0 H4: x (x y) = x ; (x y) y = x y _ _ _ H5: (x y) z = (x z) (y z) ; z (x y) = (z x) (z y): _ _ ^ _ 6. Rough graphs and their algebraic structure 115

Operation is called pseudodi erence and x y is said to be pseudodi erence of x and y.

Pomykala [49] introduced the operations of union, intersection and pseudocom- plementations over rough sets and proved that the family of rough sets with these operations form a double Stone algebra.

Theorem 6.3.9 Let (U, R) be an equivalence approximation space, and let the family of rough sets R (U) = X; X X U be equipped with the operations ; ; ; R j ^ _ : de ned in the following way:

X; X Y ; Y = X Y ; X Y ; ^ \ \ X; X Y ; Y = X Y ; X Y ; _ [ [ c c X; X = X ; X ; : X; X = (Xc;Xc): Then the structure ( R (U) ; ; ; ; ; (; ) ; (U; U)) is a complete and atomic double R ^ _ : ; ; Stone algebra.

Pagliani [45] observed that the algebra of rough sets of any equivalence approximation space is a double Heyting algebra. More formally, he proved the following result.

Theorem 6.3.10 Let ( R (U) ; ; ; ; ; ( ; ) ; (U; U)) be the algebra of rough sets of R ^ _ : ; ; an approximation space (U; R). Then, the structure ( R (U) ; ; ; ; ; ( ; ) ; (U; U)) R ^ _ ! ; ; with operations ; de ned for any X; X ; Y ; Y R (U) as follows: ! 2 R X; X Y ; Y = Z; Z ; ! 0 1 Z I(X;Y ) Z I(X;Y ) 2 [ 2 [ @ A X; X Y ; Y = Z; Z ; 0 1 Z D(X;Y ) Z D(X;Y ) 2 \ 2 \ where @ A I (X;Y ) = Z U X; X Z; Z Y ; Y ; j ^ D (X;Y ) = Z U X; X Y ; Y Z; Z ; j _ is a complete and atomic double Heyting algebra that satis es the Stone identity ( x x) = 1: : _ :: 6. Rough graphs and their algebraic structure 116

Let R(G) = (L; L) L is a subgraph of G be the family of rough graphs over R j graph G and (G) be the collection of all subgraphs of G. Following the approach of P Pomykala [49] and Pagliani [45], we'll explore algebraic structure of rough graphs. To this end, we'll start by developing the required operations. Since complement of a subgraph may not be a subgraph, two distinct operators, namely, pseudocomplement and dual pseudocomplement are in literature [62] that can serve well to ll the gap.

De nition 6.3.11 Pseudocomplement of a subgraph A, denoted by A, is the largest : subgraph disjoint from A. The dual pseudocomplement of A, denoted by A is the smallest subgraph that is necessary to add to A to get the whole graph.

Example 6.3.12 Consider the subgraphs K and L in Fig. 6.5 of Example 6.2.8. Using the above de nition, their pseudocomplements and dual pseudocomplements have been calculated with respect to G (Fig. 6.2) in Fig. 6.8 below.

n1 e1 n2 n K n1 L 5 : :

n1 e1 n2 n e3 n4 e4 n5 K 3

n1 e1 n2 e2 n3 n4 e4 n5 L Fig. 6.8. Pseudocomplement and dual pseudocomplement of K and L

Lemma 6.3.13 For K1;K2 (G), if K1 = K1, then 2 P K1 K2 = K1 K2 [ [ and

K1 K2 = K1 K2: [ [

Proof. K1 K2 K1 K2 holds by Theorem 6.2.6. For the reverse containment, [ [ let u K1 K2: By De nition 6.2.1, u K1 K2 and so u K1 K2 due to 2 [ p q [ 2 [ re exivity of R. Thus, u K1 or u K2: 2 2 If u K1; then u K1 K1 K2: 2 2 [ If u K1; then since K1 is de nable, u K1 = : Thus, u K2; that is, 62 p q \ 6 ; p q u K2: 2 Hence, in both the cases, K1 K2 K1 K2 and so K1 K2 = K1 K2: [ [ [ [ 6. Rough graphs and their algebraic structure 117

De nition 6.3.14 Let (Ki; Ki) i be a subcollection of R(G). We de ne union j 2 I G ( ) and intersection ( ) of these rough graphs as below: _ ^

(Ki; Ki) = Ki; Ki i i i ! _2I [2I [2I

(Ki; Ki) = (Mj; Mj):(Mj; Mj) (Li; Li) for all i ; 2 I i j ^2I _2J n o where the containment is de ned as: (L; L) (K; K) if and only if L K and L K: Clearly, the containment is a partial order relation. The validity of these operations have been shown in the next result.

Theorem 6.3.15 The union and intersection of a collection (Ki; Ki) i of rough j 2 I graphs is again a rough graph.

Proof. Let K = Ki S( Ki); where S( Ki) is the minimal subgraph of [ i i i [ [ [ Ki generating Ki: We claim that K = Ki and K = Ki: By Theorem 6.2.6, i i i i [ [ [ [ Ki = Ki for each i ; thus by Lemma 6.3.13 2 I

Ki = Ki = Ki: i i i [ [ [ Using the same lemma again, we get

Ki S( Ki) = Ki S( Ki) [ [ i i i i [ [ [ [ = Ki S( Ki) [ i i [ [

Now it remains to prove that S( Ki) Ki: i i [ [ Let u S( Ki); that is, u S( Ki): Since S( Ki) is a subgraph of Ki; 2 p q i i i i [ [ [ [

u Ki: p q i [

Also, since u S( Ki); by minimality of S( Ki); u will be a single node p q p q i i and a single edge class (with[ its incident nodes) if u[is an edge because otherwise if 6. Rough graphs and their algebraic structure 118 there exist some other element v not incident with u in u ; then v u and p q p q p q S( Ki) v will be minimal which contradicts minimality of S( Ki): p q i i [Thus, [ u Ki p q i [ = u Ki for some i ) p q = u Ki Ki ) 2 i [ = K = Ki S( Ki) = Ki: ) [ i i i [ [ [

Clearly, S( Ki) = Ki: Using this fact and Theorem 6.2.6 we get i i [ [

K = Ki S( Ki) [ i i [ [ = Ki S( Ki) [ i i [ [ = Ki Ki [ i ! i ! [ [ = Ki: i [ Thus, union of rough graphs will again be a rough graph. The case of intersection follows from the validity of union.

Theorem 6.3.16 ( R(G); ; ; ( ; ); (G; G)) becomes a lattice. G ; ; W V Proof. Straightforward. The following result shows that the implication operator de ned by Pagliani ! [45] for rough sets using equivalence relation is also well de ned for rough graphs using graphical equivalence relation instead of (set theoretic) equivalence relation.

Theorem 6.3.17 The structure ( R(G); ; ; ; ( ; ); (G; G)) with operation G ^ _ ! ; ; ! de ned for any (K1; K1); (K2; K2) R(G) as follows: 2 G

(K1; K1) (K2; K2) = K; K ! 0 1 K I(K1;K2) K I(K1;K2) 2 [ 2 [ @ A where

I(K1;K2) = K G (K1; K1) (K; K) (K2; K2) 2 P j ^ is a Heyting algebra. 6. Rough graphs and their algebraic structure 119

Proof. From Theorem 6.3.16, ( R(G); ; ; ; ( ; ); (G; G)) is a lattice. By the G ^ _ ! ; ; de nition of ; it is clear that (K1; K1) (K2; K2) is the greatest subgraph K of ! ! G such that

(K1; K1) K (K2; K2): ^ Thus, by Lemma 6.3.7, ( R(G); ; ; ; ( ; ); (G; G)) is a Heyting algebra. G ^ _ ! ; ; In a Heyting algebra, pseudocomplement is characterized using implication operator as follows: For any K G; K = K : 2 P : ! ; In case of rough graphs, implication operator de ned in Theorem 6.3.17 can be ! characterized as follows.

Theorem 6.3.18 For any (L; L) R(G); (L; L) = (L; L) ( ; ) is the greatest 2 G : ! ; ; element of R(G) disjoint from (L; L): G Proof. (L; L) = (M; M) : M I(L; ) 2_ ; where I(L; ) = (M; M) (M; M) (L; L) = ( ; ) : ; f j ^ ; ; g Clearly, (L; L) R(G) and ( ; ) is a lower bound of (L; L); (L; L) : (L; L) : 2 G ; ; f : g will be disjoint from (L; L) if ( ; ) is the greatest lower bound of (L; L); (L; L) : : ; ; f : g Let (K; K) be another lower bound of (L; L); (L; L) such that K is non-empty. f : g This implies that (K; K) (L; L) (K; K) (M; M) M I(L; ) 2_ ; That is, K L; K L; K M and K M: By hypothesis, M I(L; ) M I(L; ) 2[ ; 2[ ; K * M for any M I(L; ): This implies that there exist some x K such that 2 ; 2 x M; that is, 2 x = u for all u M: (6.1) 2 p q 2 But then, since K M = M; M I(L; ) M I(L; ) 2[ ; 2[ ; x v for some v M 2 p q 2 M I(L; ) 2[ ; or x v for some v M such that M I(L; ) 2 p q 2 2 ; which contradicts 6.1. Thus, (L; L) is disjoint from (L; L): Also, it is greatest, : being the union of all the elements in R(G) disjoint from (L; L): G 6. Rough graphs and their algebraic structure 120

Unlike rough set algebra, rough graph algebra does not form a Stone algebra because Stone identity is not satis ed as the following example justi es.

Example 6.3.19 Let G be the graph equipped with graphical equivalence relation R as shown in Fig. 6.2 of Example 6.2.3. The arrows relating every element to itself and every edge to its incident nodes have been omitted and it is assumed that R is re exive. Consider the subgraph L = n4; e4; n5 of G. f g Using Theorem 6.3.18;

(L; L) = ( n1; e1; n2; e2; n3; e3; n4 ; n1; e1; n2; e2; n3; e3; n4 ) : f g f g and

(L; L) = ( n5 ; n2; e2; n3; n4; e4; n5 ): :: f g f g But

(L; L) (L; L) = ( n1; e1; n2; e2; n3; e3; n4; n5 ; n1; e1; n2; e2; n3; e3; n4; e4; n5 ) : _ :: f g f g = (G; G): 6

6.4 Conclusion

In this chapter we de ne rough approximations of graphs and call the pair of lower and upper approximation as rough graph. We explore the properties of these approximation operators. It has been proved that the collection of all the rough graphs over a graph forms a lattice and more speci cally a Heyting algebra with respect to the operations de ned in the chapter. We characterize the implication operator for rough graphs and showed that the collection of rough graphs does not form a Stone algebra as it does not satisfy the Stone identity. The generalization of rough graphs to rough hypergraphs needs further investigation. Bibliography

[1] M. I. Ali, B. Davvaz, M. Shabir, Some properties of generalized rough sets, Inform. Sci., 224 (2013) 170{179.

[2] M. P. Amiri, Project selection for oil- elds development by using the AHP and fuzzy TOPSIS methods, Expert Syst. Appl., 37 (2010) 6218{6224.

[3] M. Banerjee, S. K. Pal, Roughness of a fuzzy set, Inform. Sci., 93 (1996) 235-246.

[4] M. Behzadian, S. K. Otaghsara, M. Yazdani, J. Ignatius, A state-of the-art survey of TOPSIS applications, Expert Syst. Appl., 39 (17) (2012) 13051-13069.

[5] I. Bloch, On links between mathematical morphology and rough sets, Pattern Recognition, (33) (2000) 1487{1496.

[6] I. Bloch, Heijmans HJAM, C. Ronse, Mathematical Morphology, In: Aiello M, Pratt-Hartmann I, van Benthem J, editors. Handbook of Spatial Logics. Springer; (2007) 857{944.

[7] R. Brown, I. Morris, J. Shrimpton, C. D. Wensley, Graphs of morphisms of graphs, Electronic Journal of Combinatorics, 15 (#A1) www.combinatorics.org/ojs/ (2008).

[8] K. Chakrabarty, R. Biswas, S. Nanda, Fuzziness in rough sets, Fuzzy Sets and Systems 110 (2000) 247-251.

[9] C. L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl., 24 (1968) 182-190.

[10] Y. Cheng, Forward approximation and backward approximation in fuzzy rough sets, Neurocomputing 148 (2015) 340-353.

[11] M. De Cock, C. Cornelis, E. E. Kerre, Fuzzy rough sets: the forgotten step, IEEE Trans. Fuzzy Syst. 15(1) (2007) 121{130.

121 BIBLIOGRAPHY 122

[12] C. Cornelis, M. De Cock, A. M. Radzikowska, Fuzzy rough sets: from theory into practice, In: W. Pedrycz, A. Skowron, V. Kreinovich, (eds.) Handbook of Granular Computing, Wiley, Chichester (2008).

[13] C. Cornelis, N. Verbiest, R. Jensen, Ordered weighted average based fuzzy rough sets, in: Proceedings of the 5th International Conference on Rough Sets and Knowledge Technology, RSKT2010, (2010) 78{85.

[14] J. Cousty, L. Najman, F. Dias, J. Serra, Morphological Filtering on Graphs. Computer Vision and Image Understanding, 117 (2013) 370{385.

[15] JH. Dai, Rough 3-valued algebras, Information Sciences 178 (8) (2008) 1986-1996.

[16] L. D'eer, N. Verbiest, C. Cornelis, L. Godo, A comprehensive study of implicator{ conjunctor-based and noise-tolerant fuzzy rough sets: De nitions, properties and robustness analysis, Fuzzy Sets and Systems 275 (2015) 1-38.

[17] D. Dubois, H. Prade, Rough fuzzy sets and fuzzy rough sets, Int. J. Gen. Syst., 17 (1990) 191-209.

[18] D. Dubois, H. Prade, Putting fuzzy sets and rough sets together, in: R. S low- iTnski (Ed.), Intelligent Decision Support, Kluwer Academic, Dordrecht, (1992) 203{232.

[19] I._ Ertugrul, N. Karakasoglu,Comparison of fuzzy AHP and fuzzy TOPSIS methods for facility location selection, Int. J. Adv. Manuf. Tech., 39 (7-8) (2008) 783- 795.

[20] Q. Gang, W. Hai, F. Xiangqian, Generalized hesitant fuzzy sets and their application in decision support system, Knowl-Based Syst., 37 (2013) 357{365.

[21] G. Gediga, I. D•untsch, Rough approximation quality revisited, Artif. Intell., 132 (2) (Nov. 2001) 219{234.

[22] M. Gehrke, E. Walker, On the structure of rough sets, Bulletin of the Polish Academy of Sciences, Mathematics 40 (1992) 235-245.

[23] B. Hutton, I. Reilly, Separation axioms in fuzzy topological spaces, Fuzzy Set. Syst., 3 (1980) 93-104.

[24] C. L. Hwang, K. Yoon, Multiple attributes decision making methods and applications, Springer, Berlin (1981). BIBLIOGRAPHY 123

[25] C. Kahraman, U. Cebeci, Z. Ulukan, Multi-criteria supplier selection using fuzzy AHP, Logist. Inf. Manag., 16 (6) (2003) 382{394.

[26] T. Kaya, C. Kahraman, Multi criteria decision making in energy planning using a modi ed fuzzy TOPSIS methodology, Expert Syst. Appl., 38 (2011) 6577{6585.

[27] E. P. Klement, R. Mesiar, E. Pap, Triangular norms, Trends in Logic Studia Logica Libarary, vol. 8, Kluwer Academic Publishers, DodrWecht, 2000.

[28] G. J. Klir, B. Yuan, Fuzzy Logic: Theory and Applications, Prentice-Hall, Engle- wood CliIs, NJ, 1995.

[29] M. Kondo, On structure of generalized rough sets, Inform. Sci., 176 (2006) 589- 600.

[30] P. J. M. Van Laarhoven, W. Pedrycz, A fuzzy extension of Saaty's priority theory, Fuzzy sets Syst., 11 (1) (1983) 199-227.

[31] A. H. I. Lee, W. C. Chen, C. J. Chang, A fuzzy AHP and BSC approach for eval- uating performance of IT department in the manufacturing industry in Taiwan, Expert Syst. Appl., 34 (1) (2008) 96{107.

[32] H. Li, X. Zhou, Risk decision making based on decision-theoretic rough set: A three-way view decision model, Int. J. Comput. Int. Sys., 4 (1) (2011).

[33] D. Liang, D. Liu, Systematic studies on three-way decisions with interval-valued decision-theoretic rough sets, Inform. Sci., 276 (2014) 186-203.

[34] D. Liang, D. Liu, A Novel Risk Decision Making Based on Decision-Theoretic Rough Sets Under Hesitant Fuzzy Information, IEEE. T. FUZZY SYST., 23 (2) (2015) 237-247.

[35] H. C. Liao, Z. S. Xu, Subtraction and division operations over hesitant fuzzy sets, J. Intell. Fuzzy Syst., 27 (2014) 65-72.

[36] H. Liao, Z. Xu, X. Zeng, Distance and similarity measures for hesitant fuzzy linguistic term sets and their application in multi-criteria decision making, Inform. Sci., 271 (2014) 125-142.

[37] H. Liu, R. M. Rodrguez, A fuzzy envelope for hesitant fuzzy linguistic term set and its application to multicriteria decision making, Inform. Sci., 258 (2014) 220-238. BIBLIOGRAPHY 124

[38] F. Meng, X. Chen, Q. Zhang, Multi-attribute decision analysis under a linguistic hesitant fuzzy environment, Inform. Sci., 267 (2014) 287-305.

[39] J. Mi, W. Zhang, An axiomatic characterization of a fuzzy generalization of rough sets, Information Sciences 160 (2004) 235{249.

[40] N. N. Morsi, M. M. Yakout, Axiomatics for fuzzy rough sets, Fuzzy Sets and Systems 100 (1998) 327{342.

[41] A. Nakamura, Fuzzy rough sets, Note Multiple-Valued Logic Japan 9 (8) (1988) 1{8.

[42] L. Najman, J. Cousty, A graph-based mathematical morphology reader, Pattern Recognition Letters, 47 (2014) 3{17.

[43] Y. Ouyang, Z. Wang, H. Zhang, On fuzzy rough sets based on tolerance relations, Information Sciences 180 (2010) 532-542.

[44] S. Ovchinnikov, Similarity relations, fuzzy partitions, and fuzzy orderings, Fuzzy sets and systems 40 (1991) 107-126.

[45] P. Pagliani, Rough set theory and logicalgebraic structures. In E. Or lowska (Ed.). Incomplete information: Rough set analysis (1998) (109-192). Heidelberg, Ger- many, New York: Springer-Verlag

[46] Z. Pawlak, Rough sets, Int. J. Comp. Sci., 11 (1982) 341-356.

[47] Z. Pawlak, Rough sets theoretical aspects of reasoning about data, Kluwer Aca- demic Publisher, 1991.

[48] W. Pedrycz, Granular computing: Analysis and design of intelligent systems, CRC press, 2013.

[49] J. Pomykala, J. A. Pomykala, The Stone algebra of rough sets, Bulletin of the Polish Academy of Sciences, Mathematics 36 (7-8) (1988) 495-508.

[50] Y. Qian, J. Liang, Y. Yao, C. Dang, MGRS: A multi-granulation rough set, Information Sciences 180 (2010) 949-970.

[51] K. Qin, Z. Pei, On the topological properties of fuzzy rough sets, Fuzzy Sets and Systems 151 (2005) 601-613.

[52] A. M. Radzikowska, E. E. Kerre, A comparative study of fuzzy rough sets, Fuzzy Sets and Systems 126 (2002) 137-155. BIBLIOGRAPHY 125

[53] A. M. Radzikowska, E. E. Kerre, Fuzzy rough sets based on residuated lattices, in: J. F. Peter et al. (Eds.), Transactions on Rough Sets II, LNCS 3135 (2004) 278{296.

[54] A. Mieszkowicz- Rolka, L. Rolka, Fuzzy rough approximations of process data, Int. J. Approx. Reason., 49 (2008) 301-315.

[55] D. Ruan, E. E. Kerre, Fuzzy implication operators and generalized fuzzy method of cases, Fuzzy Sets and Systems 54 (1993) 23{37.

[56] T. L. Saaty, The analytic hierarchy process, McGraw-Hill, New York (1980).

[57] J. M. F. Salido, S. Murakami, On precision aggregation, Fuzzy Sets Syst. 139 (2003) 547{558.

[58] J. M. F. Salido, S. Murakami, Rough set analysis of a general type of fuzzy data using transitive aggregations of fuzzy similarity relations, Fuzzy Sets Syst. 139 (2003) 635{660.

[59] J. Serra, Image Analysis and Mathematical Morphology, London: Academic Press; (1982).

[60] M. Shabir, M. Irfan Ali, A. Khan, Rough S-Acts, Lobachevskii J. of Math., 29 (2) (2008) 98-109.

[61] R. S lowinski,S. Greco, B. Matarazzo, Rough-set-based decision support, Search Methodologies, Springer US, (2014) 557-609.

[62] J. G. Stell, Symmetric Heyting relation algebras with applications to hypergraphs, Journal of Logical and Algebraic Methods in Programming, (84) (2015) 440-455.

[63] J. G. Stell, Relations in Mathematical Morphology with Applications to Graphs and Rough Sets, In S. Winter et al., editors, Conference on Spatial Informa- tion Theory, COSIT 2007, volume 4736 of Lecture Notes in Computer Science, Springer Verlag, (2007) 438-454.

[64] J. G. Stell, Relations on hypergraphs, In Proc. RAMiCS 13, volume 7560 of LNCS, (2012) 326-341.

[65] J. G. Stell, Relational granularity for hypergraphs, In M. Szczuka et al., editors, RSCTC Proceedings, volume 6086 of LNAI, Berlin, (2010) 267-266, Springer.

[66] J. G. Stell, R. A. Schmidt, D. Rydeheard, A bi-intuitionistic modal logic: Founda- tions and automation, Journal of Logical and Algebraic Methods in Programming, ?(?):?, 2015. BIBLIOGRAPHY 126

[67] B. Sun, W. Ma, X. Chen, Fuzzy rough set on probabilistic approximation space over two universes and its application to emergency decision-making, Expert Sys- tems (2015).

[68] S. P. Tiwari, A. K. Srivastava, Fuzzy rough sets, fuzzy preorders and fuzzy topologies, Fuzzy Sets and Systems 210 (2013) 63-68.

[69] V. Torra, Hesitant fuzzy sets, Int. J. Intell. syst., 25 (2010) 529-539.

[70] E. Triantaphyllou, C. T. Lin, Development and evaluation of ve fuzzy multiat- tribute decision-making methods. Int. J. Approx. Reason., 14 (1996) 281{310.

[71] E. C. Tsang, C. Wang, D. Chen, C. Wu, Q. Hu, Communication between information systems using fuzzy rough sets, IEEE T. Fuzzy Syst., 21 (3) (June 2013) 527-540.

[72] C. Y. Wang, B. Q. Hu, Granular variable precision fuzzy rough sets with general fuzzy relations, Fuzzy Sets and Systems (2015).

[73] P. Wasilewski, D. Slezak, Foundations of rough sets from vagueness perspective, Rough Computing: Theories, Technologies and Applications (2007) 1-37.

[74] G. Wei, Hesitant fuzzy prioritized operators and their application to multiple attribute decision making, Knowl-Based Syst., 31 (2012) 176-182.

[75] W-Z, Wu, Y. Leung, W-X. Zhang, On generalized rough fuzzy approximation operators, Transactions on Rough Sets V, Springer Berlin Heidelberg, (2006) 263-284.

[76] W. Wu, J. Mi, W. Zhang, Generalized fuzzy rough sets, Information Sciences 151 (2003) 263-282.

[77] M. M. Xia, Z. S. Xu, Hesitant fuzzy information aggregation in decision making, Int. J. Approx. Reason., 52 (3) (2011) 395-407.

[78] Z. Xu, M. Xia, Distance and similarity measures for hesitant fuzzy sets, Inform Sci., 181 (2011) 2128-2138.

[79] Z. Xu, X. Zhang, Hesitant fuzzy multi-attribute decision making based on TOP- SIS with incomplete weight information, Knowl-based Syst 52 (2013) 53-64.

[80] S. Yamak, O. Kazanci, B.Davaz, Generalized lower and upper approximations in a ring, Inform. Sci., 180 (2010) 1759-1768. BIBLIOGRAPHY 127

[81] X. Yang, X. Song, Y. Qi, J. Yang, Constructive and axiomatic approaches to hesitant fuzzy rough set, Soft Comput., 18 (2014) 1067-1077.

[82] Y. Y. Yao, Notes on rough set approximations and associated measures, Journal of zhejiang ocean university (natural science) 29 (5) (2010) 399-410.

[83] Y. Y. Yao, T. Y. Lin, Generalization of rough sets using modal logics, Intelligent Automation & Soft Computing 2(2) (1996) 103-119.

[84] Y. Y. Yao, J. Mi, Z. Li, A novel variable precision (; ) fuzzy rough set model based on fuzzy granules, Fuzzy Sets and Systems 236 (2014) 58-72.

[85] Y. Y. Yao, S. K. M. Wong, and P. Lingras, A decision theoretic rough set model, Methodologies for Intelligent Systems 5, North-Holland, New York, 17{24 (1990).

[86] L. A. Zadeh, Fuzzy sets, Information and control 8 (1965) 267-274.

[87] S. Zhao, E. C. C. Tsang, D. Chen, The model of fuzzy variable precision rough sets, IEEE Trans. Fuzzy Syst. 17(2) (2009) 451{467.

[88] W. Zhu, Relationship between generalized rough sets based on binary relation and covering, Information Sciences 179 (2009) 210{225.

[89] B. Zhu, Z. Xu, Analytic hierarchy process-hesitant group decision making, EUR. J. OPER. RES., 239 (3) (2014) 794-801.