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A STUDY ON MULTI-FUZZINESS
Thesis submitted to
KANNUR UNIVERSITY
for the award of the degree of
DOCTOR OF PHILOSOPHY
in Mathematics
under the Faculty of Science
by
SABU SEBASTIAN
Under the supervision of
Dr.T.V.RAMAKRISHNAN
Department of Mathematical Sciences
Kannur University
Mangattuparamba, Kannur
Kerala-670567, India
June 2011
To the memory of my father
T.D.Sebastian
DEPARTMENT OF MATHEMATICAL
SCIENCES
KANNUR UNIVERSITY
Dr.T.V. Ramakrishnan
Course Director
Kannur University
June 16, 2011
CERTIFICATE
Certified that the thesis entitled A Study on Multi-fuzziness is a bona fide
record of the work done by Sri. Sabu Sebastian under my supervision and guidance
in the Department of Mathematical Sciences, Kannur University and that no part of
it has been included anywhere previously for the award of any degree or title.
Dr.T.V. Ramakrishnan
Supervising Teacher
DECLARATION
I declare that the material presented in this thesis which is based on the original
work done by me under the supervision and guidance of Dr.T.V. Ramakrishnan,
has not been submitted for the award of any Degree or Diploma in any university
and that, to the best of my knowledge and belief, it contains no material previously
published by any other person, except where due reference is made.
Kannur University Sabu Sebastian
June 16, 2011
ACKNOWLEDGEMENTS
First and foremost, all thanks and praises are due to Almighty God for granting
me the chance and the ability for the completion of this study. I would like to ex-
press my deep and sincere gratitude to my supervisor, Dr.T.V. Ramakrishnan, for
his positive attitude, patience, guidance, encouragement and devotion to my personal
development through this research experience. Furthermore I am deeply grateful to
Professor T. Thrivikraman, Visiting Professor, Department of Mathematical Sciences,
Kannur University for his review, constructive criticism and advice during the entire
period of research.
I express my sincere gratitude to the faculty members, staff and my colleagues at
the Department of Mathematical Sciences, Kannur University. I am deeply indebted
to faculty members and administrative staff of the School of Information Science and
Technology, Kannur University for providing computer lab facilities. Special thanks
are due to Sri. Binu P. Chacko for the computer assistance to prepare the images
included in this thesis.
I thank the Management, Principal and Staff of Nirmalagiri College for their
wholehearted support to my research work. Also I express my thanks to the Univer-
sity Grants Commission of India for granting Teacher Fellowship under the Faculty
Development Programme.
Finally, I owe warm thanks to my loving wife Sheeja, family members and friends
for their kind support and encouragement.
Sabu Sebastian
Contents
Contents vi
List of Figures viii
List of Tables ix
Abbreviations and Notations xi
0 Introduction 1
0.1 Motivation................................. 2
0.2 Multisets and its Fuzzifications . . . . . . . . . . . . . . . . . . . . . 2
0.2.1 Multisets ............................. 2
0.2.2 FuzzyMultisets.......................... 3
0.2.3 L-multi-fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . 4
0.2.4 General Multi-fuzzy Sets . . . . . . . . . . . . . . . . . . . . . 6
0.2.5 Blizard’s Multi-fuzzy Sets . . . . . . . . . . . . . . . . . . . . 6
0.3 OutlineoftheThesis........................... 7
1 Preliminaries 11
1.1 FuzzySets................................. 11
1.1.1 Zadeh’s Extension of Functions . . . . . . . . . . . . . . . . . 13
1.2 Lattices and Lattice Valued Mappings . . . . . . . . . . . . . . . . . 14
1.2.1 Operations on Lattices . . . . . . . . . . . . . . . . . . . . . . 16
1.2.2 L-fuzzySets............................ 19
1.3 Intuitionistic Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.4 FuzzyTopology.............................. 21
1.5 FuzzyAlgebra............................... 23
1.5.1 Fuzzy Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.5.2 Lattice Valued Lattices . . . . . . . . . . . . . . . . . . . . . . 24
1.6 FuzzyLogic ................................ 25
vi
vii
1.6.1 BasicLogic ............................ 26
1.6.2 Lukasiewicz Logic . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.6.3 G¨odelLogic ............................ 27
1.6.4 ProductLogic........................... 28
1.6.5 Algebras of Logic . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.7 RoughSets ................................ 30
2 Strong L-fuzzy Lattices 32
2.1 Introduction................................ 32
2.2 FuzzySemilattices ............................ 33
2.3 Strong Fuzzy Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3 Multi-fuzzy Sets 50
3.1 Multi-fuzzySets.............................. 50
3.2 Properties of Multi-fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . 54
3.3 Multi-fuzzy Extensions of Functions . . . . . . . . . . . . . . . . . . . 57
3.4 BridgeFunctions ............................. 59
3.4.1 Extension Based on Order Homomorphisms . . . . . . . . . . 62
3.4.2 Extensions Based on Lattice Valued Fuzzy Lattices . . . . . . 68
3.4.3 Extensions Based on Lattice Homomorphisms . . . . . . . . . 76
3.5 T,Sand MOperations ......................... 76
3.5.1 p-conjugate ............................ 79
3.6 p-complements .............................. 80
3.7 Relation Between IFS and MFS Operations . . . . . . . . . . . . . . 90
3.8 Mappings on Multi-fuzzy Spaces . . . . . . . . . . . . . . . . . . . . . 93
4 Multi-fuzzy Topology 98
4.1 Introduction................................ 98
4.2 Basic Notions of Multi-fuzzy Topology . . . . . . . . . . . . . . . . . 98
4.3 Multi-fuzzy Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.4 Multi-fuzzy Compactness . . . . . . . . . . . . . . . . . . . . . . . . . 103
5 Multi-fuzzy Subgroups 105
5.1 Introduction................................ 105
5.2 Multi-fuzzy Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.3 Normal Multi-fuzzy Subgroups . . . . . . . . . . . . . . . . . . . . . . 109
6 Multi-fuzzy Logic 113
6.1 Introduction................................ 113
6.2 Multi-fuzzyLogic............................. 114
6.3 Deductive Systems of Residuated Lattices . . . . . . . . . . . . . . . 121
6.3.1 Filters of Residuated Lattices . . . . . . . . . . . . . . . . . . 122
6.3.2 FuzzyFilter............................ 123
viii
6.3.3 Multi-fuzzy Filter . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.4 Multi-fuzzy Logic in LIA . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.4.1 Implication Between Different Value Domains . . . . . . . . . 134
7 Multi-fuzzy Sets and Intuitionistic Fuzzy Sets 138
7.1 Introduction ............................... 138
7.2 Intuitionistic Fuzzy Sets Generating Maps . . . . . . . . . . . . . . . 139
8 Some Applications of Multi-fuzzy Sets 150
8.1 Multi-fuzzy Rough Sets . . . . . . . . . . . . . . . . . . . . . . . . . . 150
8.2 Taste Recognition by Multi-fuzzy Sets . . . . . . . . . . . . . . . . . 154
8.3 Image Processing Using Multi-fuzzy Sets . . . . . . . . . . . . . . . . 154
8.3.1 How to reconstruct a colour image from three linearly
independent gray images of a picture? . . . . . . . . . . . . . 156
Concluding Remarks 162
List of Publications 167
Papers Presented in National Seminars 169
Bibliography 170
List of Figures
8.1 Gray image-1 of Bird . . . . . . . . . . . . . . . . . . . . . . . . . . 159
8.2 Gray image-2 of Bird . . . . . . . . . . . . . . . . . . . . . . . . . . 159
8.3 Gray image-3 of Bird . . . . . . . . . . . . . . . . . . . . . . . . . . 159
8.4 Original image of Bird . . . . . . . . . . . . . . . . . . . . . . . . . . 159
8.5 Reconstructed image of Bird . . . . . . . . . . . . . . . . . . . . . . 159
8.6 Cropped form ((159,356) to (175,370)) Gray image-1 (Bird) . . . . . 160
8.7 Cropped from ((159,356) to (175,370)) Gray image-2 . . . . . . . . . 160
8.8 Cropped form ((159,356) to (175,370)) Gray image-3 . . . . . . . . . 160
8.9 Cropped form ((159,356) to (175,370)) Original image (Bird) . . . . 160
8.10 Cropped form ((159,356) to (175,370)) Reconstructed image . . . . . 160
8.11 Original image (Bird) with RGB values R=a, G=b, B=c . . . . . . . 161
8.12 Modified image (Bird) with RGB values R=a, G=c, B=b . . . . . . 161
8.13 Modified image (Bird) with RGB values R=b, G=a, B=c . . . . . . 161
8.14 Modified image (Bird) with RGB values R=b, G=c, B=a . . . . . . 161
8.15 Modified image (Bird) with RGB values R=c, G=a, B=b . . . . . . 161
8.16 Modified image (Bird) with RGB values R=c, G=b, B=a . . . . . . 161
ix
xi
ABBREVIATIONS AND NOTATIONS
∀ − for all (universal quantifier)
∃ − there exists (existential quantifier)
X−summation
Y−product
∨ − join
∧ − meet
_−supremum
^−infimum
⇔ − if and only if
A, B, C −arbitrary sets (crisp/ fuzzy/ multi-fuzzy)
A=B−equality of multi-fuzzy (fuzzy/ crisp) sets
AvB−inclusion of multi-fuzzy sets
AtB−union of multi-fuzzy sets
AuB−intersection of multi-fuzzy sets
A0−interior of A
¯
A−closure of A
Acor A0−classical (standard) compliment of A
A(x) or µA(x)−membership grade of x in A
CM(x)−multiplicity of x in the multiset M
A[α], Aα−α-level (α- cut) of A
X, Y and Z−universal sets (nonempty crisp sets)
I, J and K−indexing sets
i, j and k−elements of I, J and Krespectively
L, M, N, Mi, Ljand Nk−complete lattices unless it is stated otherwise
R−the set of all real numbers
Rn−nth-dimensional Euclidean space
Nm−set of all positive integers less than or equal to m
W−set of all nonnegative integers
Z+or N−set of all positive integers unless it is stated otherwise
xii
R(A)−lower approximation of Awith respect to the relation R
R(A)−upper approximation of Awith respect to the relation R
Tc
p(A, B)−p-conjugate of T(A, B) with respect to the complement operation c
Sc
p(A, B)−p-conjugate of S(A, B) with respect to the complement operation c
Cc
p(A)−p-complement of Awith respect to the complement operation c
Y
i∈I
MX
i−set of all multi-fuzzy sets in Xwith value domain Y
i∈I
Mi
Y
j∈J
LY
j−set of all multi-fuzzy sets in Ywith value domain Y
j∈J
Lj
Y
k∈K
NZ
k−set of all multi-fuzzy sets in Zwith value domain Y
k∈K
Nk
YMX
i−short form of Y
i∈I
MX
i
YLY
j−short form of Y
j∈J
LY
j
YNZ
k−short form of Y
k∈K
NZ
k
F(X)−set of all fuzzy sets in X
MkF S(X)−set of all multi-fuzzy sets in Xof dimension k and value domain Ik
AIFSGM −Atanassov intuitionistic fuzzy sets generating map
IFS −Atanassov intuitionistic fuzzy set
MFS −multi-fuzzy set
MFR-filter −multi-fuzzy regular filter
LIA −lattice implication algebra
CHAPTER 0
Introduction
Theory of fuzzy sets [107], theory of rough sets [62], theory of multi-sets [9],
theory of semisets [87] and theory of alternative (nonstandard) sets [88] are some
of the popular generalizations of classical set theory. Among them Zadeh’s fuzzy
sets are extensively studied by a number of researchers all over the world and within
the last 45 years theory of fuzzy sets has been applied to almost all the branches of
Mathematics. Goguen’s [27] L-fuzzy Sets, Atanassov’s [3] Intuitionistic Fuzzy Sets,
Chang’s [14] Fuzzy Topological Spaces, Hajek’s [29] Metamathematics of Fuzzy Logic
and Rosenfeld’s [71, 72] Fuzzy Groups, and Fuzzy Graphs are some of the important
pioneer works in the theory of fuzzy sets. Many extensions and generalizations of
Zadeh’s fuzzy set theory are developed so far. To list a few of these: L-fuzzy sets,
intuitionistic fuzzy sets, type-2 fuzzy sets [56], interval valued fuzzy sets [75], shad-
owed sets [64], internal sets [59], rough fuzzy sets and fuzzy rough sets [22, 39, 58],
vague sets [26] and gray systems [18]. Relations between some of these extensions are
1
0.1. MOTIVATION 2
studied by several authors and we list a few of them (see [11,15, 19, 21]). This thesis
entitled ‘A Study on Multi-fuzziness’ is an introductory work on multilevel fuzziness
and multi-dimensional fuzziness. Theory of multi-fuzzy sets is a generalization of
theories of fuzzy sets, L-fuzzy sets and intuitionistic fuzzy sets.
0.1 Motivation
Characterization problems like complete colour characterization of colour images,
taste recognition of food items, decision making problems with multi aspects etc.
cannot completely be characterized by a single membership function of Zadeh’s fuzzy
sets. Some of these problems can completely be characterized by multi-membership
functions of suitable multi-fuzzy sets.
0.2 Multisets and its Fuzzifications
Before presenting the concept of multi-fuzziness, we recall similar concepts or different
concepts with the name similar to multi-fuzzy sets. We start this discussion with the
recollection of important definitions and results of multisets, and their fuzzifications.
In the literature, some of such fuzzy sets are called as multi-fuzzy sets.
0.2.1 Multisets
Let Xthe universal set, Wbe the set of all nonnegative integers and let CM:
X→Wbe a function. A multiset [9, 10] Mover the set Xis the set M=
{(x, CM(x)) : x∈X, CM(x)>0}.The value CM(x) is called the multiplicity (count)
of xin M, that is, CM(x) is the number of copies of xoccur in the multiset M. See
the following example. Let X={x1, x2, ..., xn}be a universal set. Define a multiset
0.2. MULTISETS AND ITS FUZZIFICATIONS 3
Mover Xas
M={
k1
z }| {
x1, ..., x1,
k2
z }| {
x2, ..., x2, ......,
kn
z }| {
xn, ..., xn},
where kp∈Wfor p=i, 2, ..., n. For simplicity we can write M={k1/x1, k2/x2, ..., kn/xn}.
Note that here kp=CM(xp),for p= 1,2, ..., n. Following are the Miyamoto’s [53,54]
definitions of inclusion, equality, union and intersection of multisets. Let Mand N
be the multisets over X, then:
•Inclusion: M⊆Nif and only if CM(x)≤CN(x),∀x∈X;
•Equality: M=Nif and only if CM(x) = CN(x),∀x∈X;
•Union: CM∪N(x) = max{CM(x), CN(x)};
•Intersection: CM∩N(x) = min{CM(x), CN(x)}.
0.2.2 Fuzzy Multisets
In his seminal paper Yager [100] introduced the notion of Fuzzy Bags. Later
Miyamoto [52–54] renamed it as Fuzzy Multisets. In a fuzzy multiset an element of
Xmay occur more than once with possibly the same or different membership values.
See the following example. Let X={w, x, y, z}be a universal set. The set
A={(x, 0.4),(x, 0.5),(y, 0.6),(y, 1),(y, 0.6),(z, 0.7)}
is a fuzzy multiset over X. In this fuzzy multiset xoccurs two times and one xhas
the membership value 0.4 and the other has the membership value 0.5. Similarly the
membership values of yin Aare 0.6, 1 and 0.6, and the membership value of zis
0.7. From the mathematical point of view, a fuzzy multiset Ain Xis a function
A:X×[0,1] →W,where Wis the set of nonnegative integers. Usually we write the
elements of Xwith nonzero membership values only. For an x∈X, the membership
0.2. MULTISETS AND ITS FUZZIFICATIONS 4
sequence of xis defined as a non-increasing sequence of membership values of x
and it is denoted by (µ1
A(x), µ2
A(x), ..., µk
A(x)); where µ1
A(x)≥µ2
A(x)≥... ≥µk
A(x).
One can append any number of zeros at the right end of a finite sequence of the
membership values of x. It does not make any difference in occurrence of an element
x. Alternatively we can write the fuzzy multiset as:
A={(0.5,0.4)/x, (1,0.6,0.6)/y, 0.7/z}.
Miyamoto [52,55] redefined the operations: inclusion, equality, union and intersection
of fuzzy multisets. If Aand Bare the fuzzy multisets over X, then:
•Inclusion: A⊆Bif and only if µj
A(x)≤µj
B(x), j = 1,2, ..., p, ∀x∈X;
•Equality: A=Bif and only if µj
A(x) = µj
B(x), j = 1,2, ..., p, ∀x∈X;
•Union: µj
A∪B(x) = max{µj
A(x), µj
B(x)}, j = 1,2, ..., p;
•Intersection: µj
A∩B(x) = min{µj
A(x), µj
B(x)}, j = 1,2, ..., p.
The theory and applications of fuzzy multisets have been studied by several re-
searchers. We list a few of them [40,41,68–70, 73, 100].
0.2.3 L-multi-fuzzy Sets
“From a mathematical point of view, fuzzy multisets (Yager’s fuzzy bags)
are just multisets of pairs, where the first part of each pair is an element
of the universal set and the second part the degree to which the first part
belongs to the fuzzy multisets. Practically, this means that fuzzy multisets
are not fuzzy enough.”
-A. Syropoulos [80]
0.2. MULTISETS AND ITS FUZZIFICATIONS 5
Syropoulos [80] defined L-multi-fuzzy sets as a fuzzification of multisets. Let A
be a universal set, Lbe a frame and Nbe the set of all non-negative integers. An
L-multi-fuzzy set is a function M:A→L×N.If L= [0,1],he called the function
as multi-fuzzy set.
He proposed another definition for multi-fuzzy sets [79]: Assume that M:A→
Ncharacterizes a multiset M, then a multi-fuzzy set of Mis a structure Hthat
is, characterized by a function H:A→N×[0,1],such that if M(x) = n, then
H(x) = (n, i),for every x∈A. In addition, the expression H(x) = (n, i) denotes the
degree to which these ncopies of xbelongs to His i.
In both definitions multi-fuzzy sets are obtained by the fuzzification of multisets.
“Naturally, a function H:A→[0,1] ×Nis a generalization of the notion
of multisets and one that is in the spirit of the theory of fuzzy sets. Since
the term Fuzzy Multisets has been used for the structures I have described
above, I have decided to call these new structures Multi-fuzzy Sets.”
-A. Syropoulos [80]
He defined union and intersection [79] as follows: Let H, G :A→L×Nbe two
L-multi-fuzzy sets, H∪Gand H∩Gbe the union and intersection of H, G. The
membership function can be defined as follows
(H∪G)(x) = (max{Hµ(x), Gµ(x)},max{Hm(x), Gm(x)})
and
(H∩G)(x) = (min{Hµ(x), Gµ(x)},min{Hm(x), Gm(x)}),
where Hµ(x) and Gµ(x) are the membership values of xin Hand Grespectively.
Similarly Hm(x) and Gm(x) are the multiplicities of xin Hand Grespectively.
0.2. MULTISETS AND ITS FUZZIFICATIONS 6
0.2.4 General Multi-fuzzy Sets
Obtulowicz’s general multi-fuzzy sets [60] over a universal set Xare functions
M:X×N→[0,1] or equivalently, functions M:X→[0,1]N,where Nis the set of
all natural numbers. The value M(x, n) or M(x)(n) is the degree of certainty that n
copies of an object x∈Xoccur in a system or its part. In the general multi-fuzzy
sets the order relations and operations are defined component wise.
0.2.5 Blizard’s Multi-fuzzy Sets
In 1989, Blizard [9] proposed a notion of multi-fuzzy sets in the form of fuzzy sets
with nonnegative and real valued membership functions. That is, membership func-
tion of a Blizard’s multi-fuzzy set µ(x)∈[0,∞).He extended the value domain of
membership functions into the set of all real numbers and called it as general sets.
Finally, we recollect some other concepts with the name similar to multi-fuzzy.
Multi-dimensional fuzzy sets (multi-dimension extension of fuzzy numbers) of Yoshide
and Kerre [105] is an extension of fuzzy numbers. That is, a multi-dimensional fuzzy
set is a convex fuzzy set on Rnwith membership function from Rninto [0,1] is upper
semi continuous, normal and has a compact 0-cut. A multi-dimension fuzzy decision
support strategy for multi-objective and multi-layer fuzzy decision support systems is
proposed by Jiang and Chenet [32]. Subsequently, Azadi et al. [6] conducted a study
on two or more fuzzy models based on expert’s knowledge and called as multi-fuzzy
models. These three notions are fuzzy concepts. Our theory of multi-fuzzy sets deals
with the multi level fuzziness and multi-dimensional fuzziness. Roughly speaking our
multi-fuzzy sets can be characterized by multi-membership functions but they are not
fuzzifications of multisets. Our concept is more general than all the above mentioned
multi-fuzzy concepts.
0.3. OUTLINE OF THE THESIS 7
0.3 Outline of the Thesis
This thesis is divided into nine chapters, including the zero chapter. This chap-
ter (chapter 0) provides a brief introduction to the theory of multi-fuzzy sets and
multi-fuzzy logic. A brief survey of the similar concepts or different concepts with
the name similar to multi-fuzzy sets is also presented.
Chapter wise Summary
Chapter 1: This chapter presents a short summary of elementary notions and nota-
tions on fuzzy sets, L-fuzzy sets, Atanassov intuitionistic fuzzy sets, Zadeh’s extension
of crisp functions, L-fuzzy lattices [81], order homomorphisms [89], fuzzy subgroups,
fuzzy topology, basic logic, fuzzy logic, algebras of logic [12, 29, 94] and residuated
lattices [20].
Chapter 2: The notion of L-fuzzy lattice was introduced by Tepavˇcevi´c and Tra-
jkovski [81]. In this chapter we propose the concepts of L-fuzzy meet (join)-semilattice
and strong L-fuzzy lattice. We prove that join or meet of arbitrary collection of
L-fuzzy lattices and strong L-fuzzy lattices are L-fuzzy lattices and strong L-fuzzy
lattices respectively. We derive a structural hierarchy of the concepts of L-fuzzy meet
(join)-semilattice, L-fuzzy lattice, strong L-fuzzy lattice and lattice homomorphism.
Chapter 3: This chapter introduces the concepts of multi-fuzzy sets, union and
intersection of multi-fuzzy sets, bridge functions, multi-fuzzy extension of crisp func-
tions, Tand Soperations, p-conjugates of Tand Soperations, p-complements of
multi-fuzzy sets and multi-fuzzy mappings. We study the properties of multi-fuzzy
extensions of crisp functions using various bridge functions. The multi-fuzzy exten-
sions based on order homomorphisms [89], lattice homomorphisms [8], lattice valued
fuzzy lattices [81] and strong lattice valued fuzzy lattices are studied. It shows that
0.3. OUTLINE OF THE THESIS 8
the extension based on order homomorphism preserves most of the properties of
Zadeh’s fuzzy extension [107] of crisp functions. We prove that if the bridge func-
tion is an order homomorphism/ L-fuzzy lattice/ strong L-fuzzy lattice, then the
multi-fuzzy extension of a crisp function itself is an order homomorphism/ L-fuzzy
lattice/ strong L-fuzzy lattice respectively. After that we propose the notion of T
and Soperations and study some properties of such operations. Also we prove that
Tand Soperations are commutative, associative and non-decreasing operations with
identity elements. A section of this chapter is devoted for deriving relations obtained
by various compositions of p-conjugate and p-complement operations. Another sec-
tion deals with multi-fuzzy mappings and we derive some relations in multi-fuzzy
mappings. In the last sections of this chapter we discuss the interrelationship be-
tween Atanassov intuitionistic fuzzy operations [5] and multi-fuzzy operations like
p-conjugates, p-complements and multi-fuzzy mappings.
Chapter 4: The concepts of multi-fuzzy topology, multi-fuzzy continuity and
multi-fuzzy compactness are introduced in this chapter. Multi-fuzzy topology and
multi-fuzzy continuity are defined in the similar fashion of fuzzy topology and fuzzy
continuity by Chang [14]. We derive some equivalent conditions of multi-fuzzy con-
tinuity. In the last section we introduce the notion of multi-fuzzy compactness and
prove that, the image of a compact multi-fuzzy set under a multi-fuzzy continuous
function is multi-fuzzy compact.
Chapter 5: This chapter deals with the concept of multi-fuzzy subgroup. In the
pioneer paper ‘Fuzzy Groups’ Rosenfeld [71] started fuzzification of algebraic struc-
tures. In a similar manner, we define multi-fuzzy subgroup. We generalize some
results in the theory of fuzzy subgroups into multi-fuzzy subgroups. Also we define
0.3. OUTLINE OF THE THESIS 9
normal multi-fuzzy subgroups and derive some equivalent conditions for the same.
The relation between normal multi-fuzzy subgroups and normal subgroups of a group
are established using α-level sets of the multi-fuzzy subgroups. In the last section of
this chapter we prove that the image of a normal multi-fuzzy subgroup is a normal
multi-fuzzy subgroup under a multi-fuzzy extension of a group homomorphism with
respect to meet preserving order homomorphism as the bridge function. Inverses of
such extensions also preserve the normal structure of multi-fuzzy subgroups.
Chapter 6: This chapter deals with multi-fuzzy logic on residuated lattices and
BL-algebra. The notion of residuated lattice was proposed by Dilworth and Ward [20],
and BL-algebra by Hajek [29]. We propose the notion of multi-fuzzy logic in Hajek’s
sense and use multi-fuzzy sets as evaluation functions instead of fuzzy sets in the
evaluation of logical propositions. Some basic results of multi-fuzzy logic are ob-
tained and some deductive systems of residuated lattices are investigated into by us-
ing multi-fuzzy sets. For this we introduce the concept of multi-fuzzy filter and study
the properties of some classes of multi-fuzzy filters in residuated lattices. Theory of
multi-fuzzy filters is parallel to the theory of fuzzy filters by Hajek [29], Zhu [111],
Jun et al. [33–36] etc. Further we define multi-fuzzy logic in lattice implication alge-
bra and study some properties of implication relation between different value domains.
Chapter 7: The concept of Atanassov intuitionistic fuzzy sets generating maps
(AIFSGM) is introduced and studied in this chapter. AIFSGM is a multi-fuzzy
mapping producing Atanassov intuitionistic fuzzy sets from multi-fuzzy sets. We in-
vestigate into some of such functions and derive some basic results.
Chapter 8: In this chapter we characterize some practical problems using multi-
0.3. OUTLINE OF THE THESIS 10
membership functions, which cannot be characterized by a single fuzzy membership
function. We propose an approximation technique using the theories of multi-fuzzy
sets and Pawlak’s rough sets [62] and it is denoted by multi-fuzzy rough sets. We
propose complete characterization of taste recognition of food by 5-dimensional multi-
membership function obtained from the five basic tastes: sweet, sour, salty, bitter and
umami [7, 101]. Characterization of colour images is another application of multi-
membership functions. We prove that one can reconstruct a colour image from three
gray images of a picture obtained by three linearly independent equations. This
multi-fuzzy technique is useful for some of the image processing problems.
Finally, ‘Concluding Remarks’ is followed by the list of publications, the list of
papers presented in seminars and the bibliography consisting of papers and books
mentioned in this thesis.
∞
CHAPTER 1
Preliminaries
In this chapter we present the background needed for the study of multi-fuzzy
sets. We develop the theory of multi-fuzzy sets on the platform of fuzzy set theory.
The knowledge about the developments of fuzzy sets is sufficient as a prerequisite and
so we recall some basic definitions and results consisting of fuzzy sets, intuitionistic
fuzzy sets, L-fuzzy sets, order homomorphisms, lattice valued lattices, fuzzy topology,
fuzzy groups and fuzzy logic.
1.1 Fuzzy Sets
In Cantor’s set theory, a set is defined uniquely by its elements; an element of the
universe is either in or outside the set. That is, the membership function of a set
(crisp set) assigns a value of either 1 or 0 to each element in the universe. Zadeh [107]
extended the range of membership functions into the closed interval [0,1].
Definition 1.1.1. [107] Let Xbe a nonempty set. A fuzzy set Aof Xis a mapping
11
1.1. FUZZY SETS 12
A:X→[0,1],that is,
A={(x, µA(x)) : µA(x) is the grade of membership of xin A, x ∈X}.
The set of all the fuzzy sets on Xis denoted by F(X).
Let Aand Bbe fuzzy sets on a universal set X, with the grade of membership of
xin Aand Bdenoted by µAand µBrespectively. Zadeh [107] defined the following
relations and operations:
•A=B⇔µA(x) = µB(x),∀x∈X;
•A⊆B⇔µA(x)≤µB(x),∀x∈X;
•µA∪B(x) = max{µA(x), µB(x)},∀x∈X;
•µA∩B(x) = min{µA(x), µB(x)},∀x∈X;
•µA0(x)=1−µA(x),∀x∈X, where A0is the standard fuzzy complement of A.
Definition 1.1.2. [49] A function t: [0,1] ×[0,1] →[0,1] is a t-norm if ∀a, b, c ∈
[0,1]:
(1) t(a, 1) = a;
(2) t(a, b) = t(b, a);
(3) t(a, t(b, c)) = t(t(a, b), c);
(4) b≤cimplies t(a, b)≤t(a, c).
Similarly, a t-conorm (s-norm) is a commutative, associative and non-decreasing map-
ping s: [0,1] ×[0,1] →[0,1] that satisfies the boundary condition:
1.1. FUZZY SETS 13
s(a, 0) = a, for all a∈[0,1].
Definition 1.1.3. [38] A function c: [0,1] →[0,1] is called a complement (fuzzy)
operation, if it satisfies the following conditions:
(1) c(0) = 1 and c(1) = 0,
(2) for all a, b ∈[0,1], if a≤b, then c(a)≥c(b).
Definition 1.1.4. [38] A t-norm tand a t-conorm sare dual with respect to a fuzzy
complement operation cif and only if
c(t(a, b)) = s(c(a), c(b))
and
c(s(a, b)) = t(c(a), c(b)),
for all a, b ∈[0,1].
Definition 1.1.5. [38] Let nbe an integer greater than or equal to 2. A function
h: [0,1]n→[0,1] is said to be an aggregation operation for fuzzy sets, if it satisfies
the following conditions:
(1) his continuous;
(2) his monotonic increasing in all its arguments;
(3) h(0,0, ..., 0) = 0;
(4) h(1,1, ..., 1) = 1.
1.1.1 Zadeh’s Extension of Functions
For the sake of simplicity we will use A(x) and f(A)(y) instead of µA(x) and µf(A)(y)
respectively.
1.2. LATTICES AND LATTICE VALUED MAPPINGS 14
Definition 1.1.6. Let f:X→Ybe a crisp function. The fuzzy extension of f
and the inverse of the extension are f:F(X)→ F(Y) and f−1:F(Y)→ F (X)
defined by
f(A)(y) = _
y=f(x)
A(x), A ∈ F (X), y ∈Y
and
f−1(B)(x) = B(f(x)), B ∈ F (Y), x ∈X.
Theorem 1.1.7. [14] Let fbe a function from Xto Y, then:
(1) (f−1(B))0=f−1(B0),for any fuzzy set Bin Y;
(2) (f(A))0⊆f(A0),for any fuzzy set Ain X;
(3) B1⊆B2implies f−1(B1)⊆f−1(B2),where B1, B2are fuzzy sets in Y;
(4) A1⊆A2implies f(A1)vf(A2),where A1, A2are fuzzy sets in X;
(5) A⊆f−1(f(A)),for any fuzzy set Ain X;
(6) f(f−1(B)) ⊆B, for any fuzzy set Bin Y.
1.2 Lattices and Lattice Valued Mappings
One of the important concepts in all of mathematics is that of a relation. Among
various kinds of relations, equivalence relations, functions and order relations have
major role in our study. Here we concentrate on the latter concept.
Definition 1.2.1. (See [8]) A partially ordered set (or poset) is a set in which a
binary relation x≤yis defined, which satisfies for all x, y, z the following conditions:
P1. For all x, x≤x. (Reflexivity)
1.2. LATTICES AND LATTICE VALUED MAPPINGS 15
P2. If x≤yand y≤x, then x=y. (Antisymmetry)
P3. If x≤yand y≤z, then x≤z. (Transitivity)
Definition 1.2.2. (See [8]) A mapping ffrom a poset Pinto a poset Qis called
order preserving, if x≤yimplies f(x)≤f(y).A mapping gfrom Pinto Qis called
an order reversing function (antitone) if and only if x≤yimplies g(y)≤g(x).
Definition 1.2.3. (See [8]) A lattice is a partially ordered set in which x∧y=
inf(x, y) and x∨y= sup(x, y) exist for any pair of elements xand y. A sublattice of
a lattice Lis a subset Xof Lsuch that a, b ∈Ximplies a∧b∈Xand a∨b∈X.
A lattice Lis complete when each of its subsets Xhas a l.u.b (sup) and a g.l.b (inf)
in L. A lattice Lis said to be distributive, if x∧(y∨z) = (x∧y)∨(x∧z) and
x∨(y∨z) = (x∨y)∧(x∨z),for every x, y, z ∈L.
A lattice Lis said to have a lower bound 0L,if 0L≤x, ∀x∈L. Analogously, Lis
said to have an upper bound 1L,if x≤1L∀x∈L. We say Lis bounded, if Lhas both
a lower bound 0Land an upper bound 1L.In such a lattice we have the identities
0L∧x= 0L,0L∨x=x, 1L∧x=xand 1L∨x= 1L.Any finite lattice is bounded
as well as complete. An element a∈Lis called a complement of an element b∈L, if
a∧b= 0Land a∨b= 1L.A lattice Lis said to be complemented if Lis bounded and
every element in Lhas a complement. In a bounded distributive lattice, complements
are unique, if they exist.
Definition 1.2.4. (See [104]) A complete lattice Lis called infinitely distributive,
if it satisfies the conditions:
a∧_B=_
b∈B
(a∧b) and a∨^B=^
b∈B
(a∨b),∀a∈L, ∀B⊆L.
Proposition 1.2.5. (See [104]) A complete lattice Lis infinitely distributive if and
1.2. LATTICES AND LATTICE VALUED MAPPINGS 16
only if
_A∧_B=_
a∈A,b∈B
(a∧b) and ^A∨^B=^
a∈A,b∈B
(a∨b),∀A, B ⊆L.
Proposition 1.2.6. (See [8]) In any poset P, the operations of meet and join satisfy
the following laws, whenever the expressions exist:
L1. x∧x=x, x ∨x=x. (Idempotent)
L2. x∧y=y∧x, x ∨y=y∨x. (Commutative)
L3. x∧(y∧z) = (x∧y)∧z, x ∨(y∨z) = (x∨y)∨z. (Associative)
L4. x∧(x∨y) = x∨(x∧y) = x. (Absorption)
Moreover x≤yis equivalent to each of the conditions:
x∧y=xand x∨y=y. (Consistency)
Note 1.2.7. (See [8]) A semilattice is a set Lwith a binary operation ’∗’ which
is idempotent, commutative and associative. Let Pbe any poset in which any two
elements have a meet. Then Pis a semilattice with respect to the binary operation
∧.Such semilattices are called meet-semilattices. Join-semilattices are defined in a
similar manner. Any system Lwith two binary operations which satisfy the conditions
L1, L2, L3 and L4 is a lattice, and conversely.
1.2.1 Operations on Lattices
Definition 1.2.8. (See [104]) If {Lj:j∈J}is a family of lattices, then the product
Y
j∈J
Ljis a lattice if for arbitrary x, y ∈Y
j∈J
Lj, the join x∨yand the meet x∧yof
x, y are defined as for every j∈Jand for every xj, yj∈Lj:
(x∨y)j=xj∨yj
1.2. LATTICES AND LATTICE VALUED MAPPINGS 17
and
(x∧y)j=xj∧yj
or, equivalently, x≤yis defined by
xj≤jyj,∀j∈J,
where ≤and ≤jare the order relations in Y
j∈J
Ljand Ljrespectively.
Proposition 1.2.9. (See [104]) Let {Lj:j∈J}be a family of posets. Then:
(1) Y
j∈J
Ljis a poset if and only if ∀j∈J, Ljis a poset;
(2) Y
j∈J
Ljis a lattice if and only if ∀j∈J, Ljis a lattice;
(3) Y
j∈J
Ljis a complete lattice if and only if ∀j∈J, Ljis a complete lattice;
(4) Y
j∈J
Ljis a distributive lattice if and only if ∀j∈J, Ljis a distributive lattice;
(5) Y
j∈J
Ljis an infinitely distributive lattice if and only if ∀j∈J, Ljis an infinitely
distributive lattice.
Definition 1.2.10. (See [8]) Let θ:L→Mbe a function from a lattice Lto a
lattice M. Then θis order preserving (isotone) when x≤yimplies θ(x)≤θ(y); a
join-morphism (join homomorphism) when
θ(x∨y) = θ(x)∨θ(y) for all x, y ∈L; (1.i)
and a meet-morphism (meet homomorphism) when
θ(x∧y) = θ(x)∧θ(y) for all x, y ∈L. (1.ii)
θis a lattice morphism (lattice homomorphism) when (1.i) and (1.ii) hold.
1.2. LATTICES AND LATTICE VALUED MAPPINGS 18
A lattice homomorphism is called: (i) an isomorphism if it is a bijection, (ii) an
epimorphism if it is onto, (iii) a monomorphism if it is one-one, (iv) an endomorphism
if L=M, (v) an automorphism if it is an isomorphism and L=M.
Definition 1.2.11. (See [104]) Let Land Mbe complete lattices and h:L→Mbe
a mapping. The map his called a complete join preserving or arbitrary join preserving
map, if for any A⊆L
h(_A) = _
x∈A
h(x); (1.iii)
a complete meet preserving or arbitrary meet preserving map if for any A⊆L,
h(^A) = ^
x∈A
h(x).(1.iv)
The map his a complete lattice homomorphism when (1.iii) and (1.iv) hold.
Definition 1.2.12. (See [104]) Let Lbe a lattice. A mapping 0:L→Lis called
an order reversing involution, if for all a, b ∈L:
(1) a≤b⇒b0≤a0;
(2) (a0)0=a.
The symbols 0and Nare used in this thesis for order reversing involutions.
Definition 1.2.13. [89] Let 0:M→Mand 0:L→Lbe order reversing involutions.
A mapping h:M→Lis called an order homomorphism, if it satisfies the conditions:
(1) h(0M) = 0L;
(2) h(∨ai) = ∨h(ai);
(3) h−1(b0) = (h−1(b))0,
where h−1:L→Mis defined by, for every b∈L,
h−1(b) = ∨{a∈M:h(a)≤b}.
1.2. LATTICES AND LATTICE VALUED MAPPINGS 19
Proposition 1.2.14. [89] If 0:M→Mand 0:L→Lare order reversing
involutions and h:M→Lis an order homomorphism, then for every a, ai∈Mand
b, bi∈L
(1) h−1(0L)=0M;
(2) h−1(1L)=1M;
(3) a1≤a2implies h(a1)≤h(a2),that is, his an order preserving map;
(4) b1≤b2implies h−1(b1)≤h−1(b2),that is, h−1is an order preserving map;
(5) a≤h−1(b) if and only if h(a)≤bif and only if h−1(b0)≤a0;
(6) h−1(∨bi) = ∨h−1(bi),that is, h−1is an arbitrary join preserving map;
(7) h−1(∧bi) = ∧h−1(bi),that is, h−1is an arbitrary meet preserving map;
(8) a≤h−1(h(a));
(9) h(h−1(b)) ≤b.
Proposition 1.2.15. [103] Let f:L1→L2be a union (join) preserving map. If f
is injective, then
f−1(f(a)) = a, ∀a∈L1
and if fis surjective, then
f(f−1(b)) = b, ∀b∈L2.
1.2.2 L-fuzzy Sets
Definition 1.2.16. [27] Let Xbe a nonempty ordinary set and Lbe a partially
ordered set. An L-fuzzy set on Xis a mapping A:X→L, that is, the family of all
the L-fuzzy sets on Xis just LXconsisting of all the mappings from Xto L.
1.3. INTUITIONISTIC FUZZY SETS 20
Equality of L-fuzzy sets and inclusion of L-fuzzy sets are defined in similar to the
respective relations on fuzzy sets. For any A, B ∈LX,the membership functions of
A∪Band A∩Bare defined as follows:
µA∪B(x) = µA(x)∨µB(x) and
µA∩B(x) = µA(x)∧µB(x),for all x∈X.
Definition 1.2.17. [38, 112] Let Xbe a nonempty ordinary set, Lbe a complete
lattice, α∈Land A∈LX.An α-level set (or α-cut) of a fuzzy set Ais a crisp set
A[α]={x∈X:α≤A(x)}.
Aαis also denoted by α-level set of the fuzzy set A.
1.3 Intuitionistic Fuzzy Sets
Throughout this thesis intuitionistic fuzzy set means Atanassov intuitionistic fuzzy
sets. It is a generalization of the notion of Zadeh’s fuzzy sets with the condition that
the sum of degrees of membership and nonmembership is less than or equal to one.
Definition 1.3.1. [3] An Intuitionistic Fuzzy Set on Xis a set
A={hx, µA(x), νA(x)i:x∈X},
where µA(x)∈[0,1] denotes the membership degree and νA(x)∈[0,1] denotes the
non-membership degree of xin Aand
µA(x) + νA(x)≤1,∀x∈X.
1.4. FUZZY TOPOLOGY 21
Definition 1.3.2. [4] Let Lbe a complete lattice with an order reversing involution
N:L→L. An intuitionistic L-fuzzy set (lattice valued intuitionistic fuzzy set) is an
object of the form
A={(x, µ1(x), µ2(x)) : x∈X},
where µ1and µ2are functions µ1:X→L,µ2:X→L, such that for all x∈X,
µ1(x)≤ N (µ2(x)).
1.4 Fuzzy Topology
Among various branches of Mathematics, Topology is one of the first subjects where
the notion of fuzzy sets was applied. Chang [14] introduced the concept of fuzzy
topology, and subsequently Lowen [45] proposed a modified definition of fuzzy topol-
ogy. Wong [92], Conrad [16] and Mira [51] studied various aspects of fuzzy topology.
Definition 1.4.1. [14] A fuzzy topology is a family τof fuzzy sets in Xwhich
satisfies the following conditions:
(1) φ, X ∈τ;
(2) If A, B ∈τ, then A∩B∈τ;
(3) If Ai∈τfor each i∈I, then [
i∈I
Ai∈τ.
Note 1.4.2. The ordered pair (X, τ ) is called a fuzzy topological space (or fts for
short). Fuzzy sets in τare called τ-open fuzzy sets in X, simply open fuzzy sets in X.
A fuzzy set A∈ F(X) is called τ-closed if and only if its complement A0is τ-open.
The collection of all constant fuzzy sets in Xis a fuzzy topology on X.
Definition 1.4.3. [45] δ⊆ F(X) is a fuzzy topology on Xif and only if:
(1) α∈δ, for every constant α∈ F(X);
1.4. FUZZY TOPOLOGY 22
(2) A∩B∈δ, for every A, B ∈δ;
(3) _
i∈I
Ai∈δ, for every Ai∈δ.
Definition 1.4.4. [14] A fuzzy set Uin a fts (X, τ) is a neighborhood of a fuzzy
set Cif and only if there exists an open fuzzy set Osuch that C⊆O⊆U. Let A
and Bbe fuzzy sets in a fts (X, τ ), and let B⊆A. Then Bis called an interior fuzzy
set of Aif and only if Ais a neighborhood of B. The union of all interior fuzzy sets
of Ais called the interior of Aand is denoted by A0.
A0is open and is the largest open fuzzy set contained in A. The fuzzy set Ais
open if and only if A=A0(see [14]). Closure of Ais the meet of all the closed subsets
containing Aand is denoted by ¯
A(see [45]).
Definition 1.4.5. [14] Let f:F(X)→ F(Y) be a fuzzy extension of f:X→Y
and f−1:F(Y)→ F(X) be the inverse of the extension. f: (X, τ )→(Y, ρ) is said
to be fuzzy continuous, if for each fuzzy set B∈ρ, then the fuzzy set f−1(B)∈τ.
Theorem 1.4.6. [14] Let (X, τ ) and (Y, ρ) be fuzzy topological spaces and let fbe
a function from Xinto Y. Then, fis fuzzy continuous if and only if f−1(C) is closed
in X, for each closed fuzzy set Cin Y.
Proposition 1.4.7. [14] If f: (X, τ)→(Y, ρ) and g: (Y, ρ)→(Z, δ) are fuzzy
continuous, then g◦f: (X, τ)→(Z, δ) is fuzzy continuous.
Definition 1.4.8. [14] A family Uof fuzzy sets is a cover of a fuzzy set Aif and
only if A⊆ ∪{U:U∈ U}.It is an open cover if and only if each member of Uis an
open fuzzy set. A subfamily of Uis called a subcover of A, if it is an open cover of A.
Definition 1.4.9. [14] A fuzzy topological space (X, τ ) is compact if and only if
each open cover has a finite subcover.
Proposition 1.4.10. [14,45] Let (X, τ ) is compact and fa fuzzy continuous map-
ping from (X, τ ) onto (Y, ρ),then (Y, ρ) is compact.
1.5. FUZZY ALGEBRA 23
1.5 Fuzzy Algebra
Fuzzy approach to algebraic concepts started with Rosenfeld’s [71] paper on fuzzy
groups. That paper led to extensive study of fuzzy subsystems of various algebraic
structures. Das [17] studied the inter-relationship between the fuzzy subgroup and
its α-level subsets. Fuzzy normal subgroups were studied by Liu [44], Wu [93], and
Mukherjee and Bhattacharya [57]. In this section we review some definitions and
results in that theory of fuzzy algebra.
1.5.1 Fuzzy Subgroups
Definition 1.5.1. [71] A fuzzy set Aof a group Gis called a fuzzy subgroup of G
if
(1) min{A(x), A(y)} ≤ A(xy), and
(2) A(x−1)≤A(x),∀x, y ∈G.
Combine the two conditions we can write min{A(x), A(y)} ≤ A(xy−1),∀x, y ∈
G. It follows immediately from this definition that A(x)≤A(e) and A(x−1) =
A(x),∀x, y ∈G, where eis the identity element of G. A fuzzy subset Aof a group G
is called a fuzzy sub-groupoid of G, if min{A(x), A(y)} ≤ A(xy),∀x, y ∈G.
Proposition 1.5.2. [71] If {Ai:i∈I}is a family of fuzzy subgroups of a group
G, then ∩Aiis a fuzzy subgroup of G. But the union of two fuzzy subgroups of G
need not be a fuzzy subgroup of G.
A fuzzy set Ain Xis said to have the sup property if, for any subset S⊆X,
there exists s0∈Ssuch that A(s0) = sups∈SA(s).
Proposition 1.5.3. (See [71]) Let G1and G2be groups, fbe a group homomorphism
from G1into G2.
1.5. FUZZY ALGEBRA 24
•Let Abe a fuzzy subgroup in G1that has the sup property. Then f(A) is a
fuzzy subgroup of G2;
•Let Bbe a fuzzy subgroup in G2. Then f−1(B) is a fuzzy subgroup of G1.
Proposition 1.5.4. [17] If Ais a fuzzy subgroup of a group G, then each α-level
subset A[α]is subgroup of G, for α∈[0,1].
Definition 1.5.5. [44, 57,93] A fuzzy subgroup Aof a group Gis called a normal
fuzzy subgroup if and only if A(xy) = A(yx),∀x, y ∈G.
Proposition 1.5.6. [57] If Ais a normal fuzzy subgroup of a group G, then each
α-level subgroups of Ais normal in G, for α∈[0,1].
Proposition 1.5.7. [2] The intersection ∩Aiof an arbitrary family of normal fuzzy
subgroups of a group G is a normal fuzzy subgroup of G.
1.5.2 Lattice Valued Lattices
Definition 1.5.8. [50] Let (M, ∨M) be a join-semilattice and (L, ∧L,∨L) be a
complete lattice with the least element 0Land the greatest element 1L. A mapping
A:M→Lis called an L-fuzzy sub-semilattice (L-fuzzy semilattice) of Mif all the
p-level sets (p∈L) of Aare sub-semilattices of M. The set of all L-fuzzy subsets of
Mis denoted by FL(M).
Proposition 1.5.9. [50] Let (M, ∨M) be a (join) semilattice and (L, ∧L,∨L) be a
complete lattice with the least element 0Land the greatest element 1L.A∈ FL(M)
is an L-fuzzy sub-semilattice of Mif and only if
A(x)∧LA(y)≤A(x∨My),∀x, y ∈M.
Definition 1.5.10. [81] Let (M, ∧M,∨M) be a lattice and Lbe a complete lattice
with the least element 0Land the greatest element 1L. The mapping A:M→Lis
1.6. FUZZY LOGIC 25
called a lattice-valued fuzzy lattice (L-fuzzy lattice) if all the p-level sets (p∈L) of
Aare sublattices of M.
Proposition 1.5.11. [81] Let A:M→Lbe an L-fuzzy lattice, and let p, q ∈L.
If p≤q, then the q-level set
Aq={x∈M:q≤A(x)}
is a sublattice of the p-level set
Ap={x∈M:p≤A(x)}.
Proposition 1.5.12. [81] Let (M, ∧M,∨M) be a lattice and (L, ∧L,∨L) a complete
lattice with 0Land 1L. Then the mapping A:M→Lis an L-fuzzy lattice if and
only if both of the following relations hold for all x, y ∈M:
(1) A(x)∧LA(y)≤A(x∧My);
(2) A(x)∧LA(y)≤A(x∨My).
1.6 Fuzzy Logic
In a classical logic system, every proposition is either true or false. That is, truth
value are either 0 or 1. The classical two-valued logic can be extended into three-
valued logic in various ways. A logic system having three or more truth values is
called many valued logic. Formal many-valued logics, which form the basis for formal
logic, were first studied by the Polish mathematician Lukasiewicz [46] in 1920. He
developed a series of many-valued logical systems, from three valued to infinite-valued.
Later Goguen [28] connected fuzzy sets with many-valued logic and proposed a formal
fuzzy logic system. In 1998 H´ajek [29] introduced an axiomatic system (Basic logic)
for fuzzy logic and found out the common features of various fuzzy logics. Lukasiewicz