2Dzitac.pdf
INTERNATIONAL JOURNAL OF COMPUTERS COMMUNICATIONS & CONTROL
Special Issue on Fuzzy Sets and Applications (Celebration of the 50th Anniversary of Fuzzy Sets)
ISSN 1841-9836, 10(6):772-788, December, 2015.
The Fuzzification of Classical Structures: A General View
I. Dzitac
Ioan Dzitac
1. Aurel Vlaicu University of Arad,
Elena Dragoi 2, RO-310330 Arad, Romania
ioan.dzitac@uav.ro
2. Agora University of Oradea,
Piata Tineretului 8, RO-485526 Oradea, Romania
rector@univagora.ro
Abstract: The aim of this survey article, dedicated to the 50th anniversary of
Zadeh’s pioneering paper "Fuzzy Sets" (1965), is to offer a unitary view to some
important spaces in fuzzy mathematics: fuzzy real line, fuzzy topological spaces,
fuzzy metric spaces, fuzzy topological vector spaces, fuzzy normed linear spaces. We
believe that this paper will be a support for future research in this field.
Keywords: Fuzzy real line, fuzzy topological spaces, fuzzy metric spaces, fuzzy
topological vector spaces, fuzzy normed linear spaces, fuzzy F-space.
1 Introduction
An introduction in the classical set theory begins, in general, in the following way: by a set
we understand a collection of objects, well individualized, such that we can decide without any
ambiguity whether a given element belongs to that set or not. What should we do when we
cannot answer this question? Can we talk about sets described in natural language such as "the
set of beautiful women" or "the set of tall men"? Although these questions are natural they
were formulated only in 1965 by Lotfi A. Zadeh. In order to give answers to these questions,
L.A. Zadeh [61] introduced the concept of fuzzy set.
We present bellow some thoughts of Lotfi A. Zadeh, remembering the beginnings and the
current impact of fuzzy sets theory.
In [63] Lotfi A. Zadeh said: "In July of 1964, I was attending a conference in New York and
was staying at the home of my parents. They were away. I had a dinner engagement but it had
to be canceled. I was alone in the apartment. My thoughts turned to the unsharpness of class
boundaries. It was at that point that the simple concept of a fuzzy set occurred to me. It did not
take me long to put my thoughts together and write a paper on the subject. This was the genesis
of fuzzy set theory. I knew that the word "fuzzy" would make the theory controversial. Knowing
how the real world functions, I submitted my paper to Information and Control because I was a
member of the Editorial Board. There was just one review-which was very lukewarm. I believe
that my paper would have been rejected if I were not on the Editorial Board. Today (20 Dec.
2010), with over 26,000 Google Scholar citations, "Fuzzy Sets"is by far the highest cited paper
in Information and Control.
My paper was a turning point in my research. Since 1965, almost all of my papers relate to
fuzzy set theory and fuzzy logic. As I expected, my 1965 paper drew a mixed reaction, partly
because the word "fuzzy" is generally used in a pejorative sense, but, more substantively, because
unsharpness of class boundaries was not considered in science and engineering. In large measure,
comments of my paper were skeptical or hostile. An exception was Japan. In 1968, I began
to receive letters from Japan expressing interest in application of fuzzy set theory to pattern
recognition. In the years which followed, in Japan fuzzy set theory and fuzzy logic became
objects of extensive research and wide-ranging application, especially in the realm of consumer
Copyright © 2006-2015 by CCC Publications
The Fuzzification of Classical Structures: A General View 773
products. A very visible application was the subway system in the city of Sendai - a fuzzy logic-
based system designed by Hitachi and Kawasaki Heavy Industry. The system began to operate
in 1987 and is considered to be a great success."
On October 2, 2015 the paper "Fuzzy Sets" has already over 58,540 citations in Google
Scholar and all Zadeh’s papers have over 151,300 citations.
"Computation with information described in natural language (NL) is closely related to
Computing with Words. NL-Computation is of intrinsic importance because much of human
knowledge is described in natural language. This is particularly true in such fields as economics,
data mining, systems engineering, risk assessment and emergency management. It is safe to
predict that as we move further into the age of machine intelligence and mechanized decision-
making, NL-Computation will grow in visibility and importance." (L.A. Zadeh, [65]).
"What is thought-provoking is that neither traditional mathematics nor standard probability
theory has the capability to deal with computational problems which are stated in a natural
language. Not having this capability, it is traditional to dismiss such problems as ill-posed. In
this perspective, perhaps the most remarkable contribution of Computing with Words (CW) is
that it opens the door to empowering of mathematics with a fascinating capability - the capability
to construct mathematical solutions of computational problems which are stated in a natural
language. The basic importance of this capability derives from the fact that much of human
knowledge, and especially world knowledge, is described in natural language. In conclusion, only
recently did I begin to realize that the formalism of CW suggests a new and challenging direction
in mathematics - mathematical solution of computational problems which are stated in a natural
language. For mathematics, this is an unexplored territory." (L.A. Zadeh, [64]).
Since then many authors have developed the theory of fuzzy set and its applications. Espe-
cially, many mathematicians tried to extend in fuzzy context classical mathematics results. The
success of the research undertaken has been demonstrated in a variety of areas such as: artificial
intelligence, computer science, quantum particle physics, control engineering, robotics and many
more. Perhaps the main reason for this rapid development is that the world that surrounds us is
full of uncertainty, the data we collect from the environment are, in general, vague and incorrect.
So the notion of fuzzy set allows us to study the degree of uncertainty mentioned above in a
purely mathematical way.
2 Fuzzy Sets
The concept of fuzzy set was introduced by L.A. Zadeh [61] in 1965.
Definition 1. [61] A fuzzy set in X is a function µ : X → [0, 1]. We denote by F(X) the family
of all fuzzy sets in X.
Remark 2. In fact µ is the membership function of a fuzzy set A of X and the value µ(x)
represents "the grade of membership" of x to fuzzy set A. But, in this paper, we adopt the
convention to identify fuzzy sets with their membership functions. This identification was first
used by J.A. Goguen [19].
Remark 3. As any subset of X can be identified with its characteristic function we remark that
fuzzy sets generalize subsets.
Definition 4. [61] Let µ, ν be fuzzy sets in X. The union of fuzzy sets µ şi ν, denoted µ∨ν, the
intersection of fuzzy sets µ şi ν, denoted µ ∧ ν, the complement of fuzzy set µ, denoted 1 − µ,
774 I. Dzitac
are fuzzy sets in X, defined by
(µ ∨ ν)(x) = max{µ(x), ν(x)} (1)
(µ ∧ ν)(x) = min{µ(x), ν(x)} (2)
C(µ)(x) = 1 − µ(x) (3)
Definition 5. The union of the fuzzy sets {µi}i∈I is defined by
(
∨
i∈I
µi
)
(x) = sup{µi(x) : i ∈ I} .
The intersection of the fuzzy sets {µi}i∈I is defined by
(
∧
i∈I
µi
)
(x) = inf{µi(x) : i ∈ I} .
Definition 6. Let α ∈ (0, 1], and let µ be a fuzzy set in X. The α-level set [µ]α is defined by
[µ]α := {x ∈ X : µ(x) ≥ α} .
The support of µ is
supp µ := {x ∈ X : µ(x) > 0} .
Definition 7. [61] Let X be a vector space over a field K (where K is R or C). A fuzzy set µ is
called convex if
µ(λx1 + (1 − λ)x2) ≥ min{µ(x1), µ(x2)} , (∀)x1, x2 ∈ X, (∀)λ ∈ [0, 1] .
The extension principle is undoubtedly one of the most important of Zadeh’s contribution in
fuzzy set theory, allowing to extend in a fuzzy context almost any mathematical concept. The
extension principle was introduced by Zadeh [61] in 1965, and then it suffered many changes:
Zadeh [62]; Jain [24]; Dubois & Prade [14]. For more details of this principle and its extensions
we refer the reader to [66], [30].
Let X = X1 ×X2 ×···×Xr and µ1, µ2, · · · , µr be fuzzy sets in X1, X2, · · · , Xr, respectively.
Let f : X → Y . The extension principle allows us to define a fuzzy set in Y by
µ(y) =
{
sup
(x1,··· ,xr)∈f−1(y)
min{µ1(x1), · · · , µr(xr)} if f
−1(y) 6= ∅
0 if f−1(y) = ∅
.
3 Fuzzy relations
It is well known that the fuzzy relations play an important role in fuzzy modeling and fuzzy
control and they also have important applications in relational databases, approximate reasoning,
preference modeling, medical diagnosis.
The concept of fuzzy relation was introduced by L.A. Zadeh in his classical paper [61].
According to L.A. Zadeh a fuzzy relation T between two nonempty sets X and Y is a fuzzy set
in X × Y , i.e. it is a mapping T : X × Y → [0, 1]. We denote by FR(X, Y ) the family of all
fuzzy relations between X and Y . For x ∈ X we denote by Tx the fuzzy set in Y defined by
Tx(y) = T(x, y). Thus, a fuzzy relation can be seen as a mapping X ∋ x 7→ Tx ∈ F(Y ), where
F(Y ) represents the family of all fuzzy sets in Y .
The Fuzzification of Classical Structures: A General View 775
Such mappings were investigated by various mathematicians under different aspects. Thus
N. Papageorgiou [46] called these mappings fuzzy multifunctions and studied the continuity of
these mappings. E. Tsiporkova, B. De Baets, E. Kerre [56, 57] called these maps fuzzy multi-
valued mappings and they defined lower and upper semi-continuous fuzzy multivalued mapping.
The relationships between these two types were studied completely. The continuity of fuzzy
multifunctions was also studied by I. Beg [6]. In papers [7, 8], I.Beg studied the linear fuzzy
multivalued operators and vector-valued fuzzy multifunctions. An application T : Rm → F(Rn)
is called a fuzzy process (see Y. Chalco-Cano, M.A. Rojas-Medar, R. Osuna-Gómez [9]).
A special attention was given to convex fuzzy processes. They were introduced by M. Matloka
[36] in 2000. Another concept of convex fuzzy process was proposed by Y. Syau, C. Low and T.
Wu [55] in 2002. A comparative study of these fuzzy convex processes was made in 2010 by D.
Qiu, F. Yang, L. Shu [47]. To avoid any confusion D. Qiu, F. Yang and L. Shu called the former
M-convex fuzzy process and the latter SLW-convex process.
In paper [41] special types of fuzzy relations on vector spaces were considered : affine fuzzy
relations, linear fuzzy relations, convex fuzzy relations, M-convex fuzzy relations. Some funda-
mental properties of fuzzy linear relations between vector spaces are considered in [43].
The domain D(T) of T is a fuzzy set in X defined by D(T)(x) := sup
y∈Y
T(x, y) (see [56]). We
note that
supp D(T) = {x ∈ X : Tx 6= ∅} = {x ∈ X : (∃)y ∈ Y such that T(x, y) > 0} .
If for all x ∈ supp D(T) there exists unique y ∈ Y such that T(x, y) > 0, then T is called fuzzy
function (or single-valued fuzzy function). In this case, we denote this unique y by T(x).
If µ ∈ F(X), then T(µ) ∈ F(Y ) is defined by T(µ)(y) := sup
x∈X
[T(x, y) ∧ µ(x)] (see [6]). In
particular, the range R(T) of T is a fuzzy set in Y defined by R(T)(y) := sup
x∈X
T(x, y) [56].
Let T ∈ FR(X, Y ), S ∈ FR(Y, Z). The composition S ◦ T ∈ FR(X, Z) (or simply ST ) is
defined by (S ◦ T)(x, z) := sup
y∈Y
[T(x, y) ∧ S(y, z)] [61].
Proposition 8. Let T ∈ FR(X, Y ), S ∈ FR(Y, Z). Then (S ◦ T)x = S(Tx), (∀)x ∈ X.
Proposition 9. The operation ” ◦ ” is associative.
The inverse (or reverse relation) T−1 of a fuzzy relation T ∈ FR(X, Y ) is a fuzzy set in Y ×X
defined by T−1(y, x) = T(x, y). It is obvious that R(T) = D(T−1) and R(T−1) = D(T). We
remark that, for µ ∈ F(Y ), we have T−1(µ)(x) = sup
y∈Y
[T−1(y, x) ∧ µ(x)] = sup
y∈Y
[T(x, y) ∧ µ(x)] .
This type of inverse is usually called lower inverse [6].
4 Fuzzy real numbers
For the concept of fuzzy real number, arithmetic operation and ordering on the set of all
fuzzy real numbers we refer the reader to the papers [13, 14, 17, 25, 26, 38, 59].
Definition 10. A fuzzy set in R, namely a mapping x : R → [0, 1], with the following properties:
1. x is convex, i.e. x(t) ≥ min{x(s), x(r)}, for s ≤ t ≤ r;
2. x is normal, i.e. (∃)t0 ∈ R : x(t0) = 1;
3. x is upper semicontinuous, i.e.
(∀)t ∈ R, (∀)α ∈ (0, 1] : x(t) < α,
(∃)δ > 0 such that |s − t| < δ ⇒ x(s) < α
776 I. Dzitac
is called a fuzzy real number. We will denote by R(I) the set of all fuzzy real numbers.
Remark 11. Let x ∈ R(I). For all α ∈ (0, 1], the α-level sets [x]α = {t : x(t) ≥ α} are closed
intervals [aα, bα], where the values aα = −∞ and bα = ∞ are admissible. When aα = −∞, the
interval [aα, bα] will be denoted by (−∞, bα].
Definition 12. A fuzzy real number x is called non-negative if x(t) = 0, (∀)t < 0. The set of all
non-negative real numbers will be denoted by R∗(I).
Remark 13. For each r ∈ R we can consider the fuzzy real number r defined by
r(t) =
{
1 if t = r
0 if t 6= r
.
These fuzzy numbers are called crisp. Thus R can be embedded in R(I).
Definition 14. [38] The arithmetic operations +,−, ·, / on R(I), are defined by:
(x + y)(t) =
∨
s∈R
min{x(s), y(t − s)}, (∀)t ∈ R (4)
(x − y)(t) =
∨
s∈R
min{x(s), y(s − t)}, (∀)t ∈ R (5)
(xy)(t) =
∨
s∈R∗
min{x(s), y(t/s)}, (∀)t ∈ R (6)
(x/y)(t) =
∨
s∈R
min{x(ts), y(s)}, (∀)t ∈ R (7)
Remark 15. Previous definitions are special cases of Zadeh’s extension principle.
Remark 16. The additive and multiplicative operations are associative and commutative with
the identities 0 and 1, where
0(t) =
{
1 if t = 0
0 if t 6= 0
, 1(t) =
{
1 if t = 1
0 if t 6= 1
.
Remark 17. It is obvious that
1. −x = 0 − x;
2. (−x)(t) = x(−t);
3. x − y = x + (−y);
4. −(x + y) = (−x) + (−y).
Definition 18. The absolute value |x| of x ∈ R(I) is defined by
|x|(t) =
{
max{x(t), x(−t)} if t ≥ 0
0 if t < 0
.
Proposition 19. [26] The equations a + x = 0 and ax = 1 have unique solutions if and only if a
is crisp.
Definition 20. [16] A partial ordering on R(I) is defined by
x ≤ y if a1α ≤ a
2
α and b
1
α ≤ b
2
α , (∀)α ∈ (0, 1] ,
where [x]α = [a
1
α, b
1
α] and [y]α = [a
2
α, b
2
α].
The Fuzzification of Classical Structures: A General View 777
Proposition 21. [26] If [aα, bα], 0 < α ≤ 1, are the α-level sets of a fuzzy real number x, then:
1. [aα1, bα1] ⊇ [aα2, bα2], (∀)0 < α1 ≤ α2;
2. [ lim
k→∞
aαk, lim
k→∞
bαk] = [aα, bα], where {αk} is an increasing sequence in (0, 1] converging to
α.
Conversely, if [aα, bα], 0 < α ≤ 1, is a family of non-empty intervals which satisfy the conditions
(1) and (2), then the family [aα, bα] represents the α-level sets of a fuzzy real number.
Remark 22. As α-level sets of a fuzzy real number is an interval, there is a debate in the
nomenclature of fuzzy real numbers. In [15], D. Dubois and H. Prade suggested to call this fuzzy
interval. They developed a different notion of fuzzy real number by considering it as a fuzzy
element of the real line.
5 Fuzzy topological spaces
From the notion of fuzzy set, to the notion of fuzzy topological space, there was one more step
to be taken. Thus, in 1968, C.L. Chang [10] introduced the notion of fuzzy topological space.
The definition is a natural translation to fuzzy sets of the ordinary definition of topological space.
Indeed, a fuzzy topology is a family T , of fuzzy sets in X, such that T is closed with respect to
arbitrary union and finite intersection and X,∅ ∈ T .
Definition 23. [10] Let X be an arbitrary set. A fuzzy topology on X is a family T ⊂ F(X)
satisfying the following axioms:
1. ∅, X ∈ T , where ∅ is characterized by the membership function µ(x) = 0, (∀)x ∈ X and X
is characterized by the membership function µ(x) = 1, (∀)x ∈ X;
2. If µ1, µ2 ∈ T , then µ1 ∧ µ2 ∈ T ;
3. If {µi}i∈I ⊂ T , then
∨
i∈I
µi ∈ T .
The pair (X,T ) will be called fuzzy topological space. The elements of T will be called open
fuzzy sets.
Definition 24. [10] Let (X,T ) be a fuzzy topological space. A fuzzy set µ1 is a neighborhood
of a fuzzy set µ2 if there exists an open fuzzy set µ such that µ2 ⊆ µ ⊆ µ1.
Theorem 5.1. Let (X,T ) be a fuzzy topological space. A fuzzy set µ is an open fuzzy set if and
only for each fuzzy set µ2 ⊆ µ, we have that µ is a neighborhood of µ2.
Definition 25. [10] Let X, Y be arbitrary sets and f : X → Y . If µ is a fuzzy set in Y , then the
inverse of µ, denoted as f−1(µ), is a fuzzy set in X defined by
f−1(µ)(x) := µ(f(x)), (∀)x ∈ X .
Conversely, if µ is a fuzzy set in X, the image of µ, denoted as f(µ), is a fuzzy sets in Y defined
by
f(µ)(y) =
{
sup
x∈f−1(y)
µ(x) if f−1(y) 6= ∅
0 if f−1(y) = ∅
.
Definition 26. A function f from a fuzzy topological space (X,T ) to a fuzzy topological space
(Y,G) is said to be fuzzy continuous if the inverse of each open fuzzy set is an open fuzzy set.
778 I. Dzitac
In 1976, R. Lowen [33] remarked that with Chang’s definition constant functions between
fuzzy topological spaces are not necessarily continuous. Thus R. Lowen suggested an alternative
and more natural definition replacing the condition X,∅ ∈ T with every constant function belong
to T .
Let (X,T ) be a topological space. We recall that a function f : X → R is said to be lower
semi-continuous if for all a ∈ R, {x ∈ X : f(x) > a} is an open set in X.
Example 27. [33] Let (X,T ) be a topological space. The lower semi-continuous fuzzy topology
on X associated with T is
ω(T ) := {µ : X → [0, 1] : µ is lower semi-continuous} .
The usual fuzzy topology on K is the lower semi-continuous fuzzy topology generated by the
usual topology of K.
Remark 28. [34] If (X,Ti)i∈I is a family of topological spaces and T is the product topology on
X =
∏
i∈I
Xi, then ω(T ) is the product of fuzzy topologies ω(Ti), i ∈ I.
Definition 29. [33] The closure and the interior of a fuzzy set µ in a fuzzy topological space
(X,T ) are defined by
µ = inf{µ1 : µ ⊆ µ1 and C(µ1) ∈ T }
◦
µ= sup{µ1 : µ1 ⊆ µ and µ1 ∈ T } .
We must note that, in paper [37], J. Michálek defined and studied another concept of fuzzy
topological space which is quite different from the classic Chang’s definition. In paper [35] it is
shown the divergences between these two types of fuzzy topological spaces.
In paper [58], it is shown that the fuzzy continuous functions can be characterized by the
closure of fuzzy sets, a subbasis of a fuzzy topology, and a fuzzy neighborhood.
In [53] a more consistent approach to the use of ideas of fuzzy mathematics in general topology
has been developed.
Definition 30. [53] A fuzzy topological space is a pair (X,T ), where X is an arbitrary set and
T : F(X) → [0, 1] is a map satisfying the following axioms:
1. T (0) = T (1) = 1;
2. T (µ1 ∧ µ2) ≥ T (µ1) ∧T (µ2), (∀)µ1, µ2 ∈ F(X);
3. T
(
∨
i∈I
µi
)
≥
∧
i∈I
T (µi), (∀){µi}i∈I ⊆ F(X).
A nice survey concerning fuzzy topological spaces was written by A.P. Shostak [54]. This
survey contains: various approaches to the definition of fuzzy topology, fundamental interrela-
tions between the categories of fuzzy topology and the category of topological spaces, the notion
of a fuzzy point, the convergence structure in fuzzy spaces, important topological properties for
fuzzy spaces etc.
6 Fuzzy metric spaces
One of the important problems concerning the fuzzy topological spaces is to obtain an ade-
quate notion of fuzzy metric space. Many authors have investigated this question, and several
notions of fuzzy metric space have been defined and studied. We mention that the concept of
The Fuzzification of Classical Structures: A General View 779
fuzzy metric was introduced by I. Kramosil and J. Michálek [9] in 1975. Their notion is equiv-
alent, in certain sense, with that of statistical metric. We note that the statistical metrics were
studied many years before, and a brief survey on them was made by B. Schweizer and A. Sklar in
paper [52]. We also note that, in 1994, A. George and P. Veeramani [18] modified the definition
of fuzzy metric in order to obtain a Hausdorff topology on a fuzzy metric space.
Definition 31. [52] A binary operation
∗ : [0, 1] × [0, 1] → [0, 1]
is called triangular norm (t-norm) if it satisfies the following condition:
1. a ∗ b = b ∗ a, (∀)a, b ∈ [0, 1];
2. a ∗ 1 = a, (∀)a ∈ [0, 1];
3. (a ∗ b) ∗ c = a ∗ (b ∗ c), (∀)a, b, c ∈ [0, 1];
4. If a ≤ c and b ≤ d, with a, b, c, d ∈ [0, 1], then a ∗ b ≤ c ∗ d.
Example 32. Three basic examples of continuous t-norms are ∧, ·,∗L, which are defined by
a ∧ b = min{a, b}, a · b = ab (usual multiplication in [0, 1]) and a ∗L b = max{a + b − 1, 0}
(the Lukasiewicz t-norm).
Definition 33. [9] The triple (X, M,∗) is said to be a fuzzy metric space if X is an arbitrary
set, ∗ is a continuous t-norm and M is a fuzzy metric, i.e. a fuzzy set in X × X × [0,∞) which
satisfies the following conditions:
(M1) M(x, y, 0) = 0, (∀)x, y ∈ X;
(M2) [M(x, y, t) = 1, (∀)t > 0] if and only if x = y;
(M3) M(x, y, t) = M(y, x, t), (∀)x, y ∈ X, (∀)t ≥ 0;
(M4) M(x, z, t + s) ≥ M(x, y, t) ∗ M(y, z, s), (∀)x, y, z ∈ X, (∀)t, s ≥ 0;
(M5) (∀)x, y ∈ X, M(x, y, ·) : [0,∞) → [0, 1] is left continuous and lim
t→∞
M(x, y, t) = 1
Remark 34. In the definition of the fuzzy metric space, I. Kramosil and J. Michálek have imposed
another condition: "M(x, y, ·) is nondecreasing, for all x, y ∈ X". M. Grabiec [12] showed that
this statement derives from the other axioms.
Indeed, for 0 < t < s, we have
M(x, y, s) ≥ M(x, x, s − t) ∗ M(x, y, t) = 1 ∗ M(x, y, t) = M(x, y, t).
Example 35. [18] Let (X, d) be a metric space. Let
Md : X × X × [0,∞), Md(x, y, t) =
{
t
t+d(x,y)
if t > 0
0 if t = 0
.
Then (X, Md,∧) is a fuzzy metric space. Md is called standard fuzzy metric.
Theorem 6.1. [18] Let (X, M,∗) be a fuzzy metric space. For x ∈ X, r ∈ (0, 1), t > 0 we define
the open ball
B(x, r, t) := {y ∈ X : M(x, y, t) > 1 − r} .
Let
TM := {T ⊂ X : x ∈ T iff (∃)t > 0, r ∈ (0, 1) : B(x, r, t) ⊆ T} .
Then TM is a topology on X.
780 I. Dzitac
Proposition 36. [18] Let (X, d) be a metric space and Md be the corresponding standard fuzzy
metric on X. Then the topology Td induced by the metric d, and the topology TMd induced by
the standard fuzzy metric Md are the same.
Definition 37. [18] Let (X, M,∗) be a fuzzy metric space and (xn) be a sequence in X. The
sequence (xn) is said to be convergent if there exists x ∈ X such that M(xn, x, t) = 1, (∀)t > 0.
In this case, x is called the limit of the sequence (xn) and we write lim
n→∞
xn = x, or xn → x.
Remark 38. [18] Let (X, M,∗) be a fuzzy metric space. A sequence (xn) is convergent to x if
and only if (xn) is convergent to x in topology TM .
Indeed,
xn → x in topology TM ⇔
⇔ (∀)r ∈ (0, 1), (∀)t > 0, (∃)n0 ∈ N : xn ∈ B(x, r, t), (∀)n ≥ n0 ⇔
⇔ (∀)r ∈ (0, 1), (∀)t > 0, (∃)n0 ∈ N : M(xn, x, t) > 1 − r, (∀)n ≥ n0 ⇔
⇔ lim
n→∞
M(xn, x, t) = 1, (∀)t > 0 .
Definition 39. [18] Let (X, M,∗) be a fuzzy metric space and (xn) be a sequence in X. The
sequence (xn) is said to be a Cauchy sequence if
(∀)r ∈ (0, 1), (∀)t > 0, (∃)n0 ∈ N : M(xn, xm, t) > 1 − r, (∀)n, m ≥ n0 .
A fuzzy metric in which every Cauchy sequence is convergent is called complete fuzzy metric
space.
Definition 40. [18] Let (X, M,∗) be a fuzzy metric space. A subset A of X is said to be fuzzy
bounded if there exist r ∈ (0, 1) and t > 0 such that M(x, y, t) > 1 − r, for all x, y ∈ A.
Remark 41. If (X, M,∗) is a fuzzy metric space induced by a metric d on X, then A ⊆ X is
fuzzy bounded if and only if A is bounded.
We say that a topological space (X,T ) is fuzzy metrizable if the topology is generated by
a fuzzy metric. V. Gregori and S. Romaguera [22] proved that a topological space is fuzzy
metrizable if and only if it is metrizable.
In paper [21], the fuzzy metric M∗(x, y, t) :=
min{x,y}+t
max{x,y}+t
and other fuzzy metrics related to
it were studied. This fuzzy metric is useful for measuring perceptual colour differences between
colour samples.
7 Fuzzy topological vector spaces
The starting point of the theory of fuzzy topological vector spaces was a series of papers of
A.K. Katsaras (see [27], [28], [29]).
Let X be a vector space over a field K (where K is R or C).
Definition 42. [27] Let µ1, µ2, · · · , µn be fuzzy sets in X. Then µ = µ1 ×µ2 ×···×µn is a fuzzy
set in Xn defined by
µ(x1, x2, · · · , xn) = µ1(x1) ∧ µ2(x2) ∧·· ·∧ µn(xn) .
Let f : Xn → X , f(x1, x2, · · · , xn) =
n
∑
k=1
xk. The fuzzy set f(µ) is called the sum of fuzzy
sets µ1, µ2, · · · , µn and it is denoted by µ1 + µ2 + · · · + µn. In fact
(µ1 + µ2 + · · · + µn)(x) = ∨{µ1(x1) ∧ µ2(x2) ∧·· ·∧ µn(xn) : x =
n
∑
k=1
xk} .
The Fuzzification of Classical Structures: A General View 781
Let µ be a fuzzy set in X and λ ∈ K. The fuzzy set λµ is the image of µ under the map
g : X → X, g(x) = λx. Thus,
(λµ)(x) =
µ
(
x
λ
)
if λ 6= 0
0 if λ = 0, x 6= 0
∨{µ(y) : y ∈ X} if λ = 0, x = 0
.
Definition 43. [28] A fuzzy topological vector space is a vector space X over K equipped with
a fuzzy topology such that the mappings
+ : X × X → X , (x, y) 7→ x + y
· : K × X → X , (λ, x) 7→ λ · x
are fuzzy continuous when K has the fuzzy usual topology and X × X and K × X have the
corresponding product fuzzy topologies.
In paper [28], the fuzzy vector topologies were characterized in terms of the corresponding
families of neighborhoods of zero.
Theorem 7.1. [29] Let X be a vector space over K, and T be a topology on X. Then (X,T ) is
a topological vector space if and only if (X, ω(T )) is a fuzzy topological vector space.
8 Fuzzy normed linear spaces
Studying fuzzy topological vector spaces, A.K. Katsaras [29], introduced in 1984 for the first
time, the notion of fuzzy norm on a linear space. In 1992, C. Felbin [17] introduced another
concept of fuzzy norm by assigning a fuzzy real number to each element of the linear space.
In 1994, S.C. Cheng and J.N. Mordeson [5] introduced another idea of fuzzy norm on a linear
space such that their corresponding fuzzy metric was of Kramosil and Michálek type. Following
S.C. Cheng and J.N. Mordeson, in 2003, T. Bag and S.K. Samanta [2] introduced a new concept
of fuzzy norm, and studied the properties of finite dimensional fuzzy normed linear spaces. A
comparative study on fuzzy norms introduced Katsaras, Felbin and Bag and Samanta was made
in paper [4]. Other approaches for fuzzy normed linear spaces can be found in [1,7,10,44,48,51,60].
Recently, S. Nădăban introduced the concepts of fuzzy pseudo-norm and fuzzy F-space [11].
Different types of fuzzy bounded linear operators and the relation between fuzzy continuity
and fuzzy boundedness were studied in [3], in the context of Bag-Samanta’s type fuzzy normed
linear spaces. The study of fuzzy continuous mappings and fuzzy bounded linear operators in
fuzzy normed linear spaces initiated by T. Bag and S.K. Samanta in [3] was continued by I.
Sadeqi and F.S. Kia [51] as well, as S. Nădăban [45] in a more general setting.
Fuzzy bounded linear operators in Felbin’s type fuzzy normed linear space were introduced by
M. Itoh and M. Cho in [23]. J.Z. Xiao and X.H. Zhu [59,60] gave a new definition for fuzzy norm
of bounded operators. In [5], different definitions of strongly fuzzy bounded linear operators
and weakly fuzzy bounded linear operators were given and a new idea of their fuzzy norm were
introduced. In [25], some properties of the space of all weakly fuzzy bounded linear operators
were studied.
In 2006, R. Saadati and J.H. Park introduced the notion of intuitionistic fuzzy Euclidean
normed space (see [49], [50]). In paper [42] some special fuzzy norms on Kn were introduced,
and in this way, fuzzy Euclidean normed spaces were obtained, . In order to introduce this
concept it is proved that the cartesian product of a finite family of fuzzy normed linear spaces is
a fuzzy normed linear space.
782 I. Dzitac
Definition 44. [27] A fuzzy set ρ in X is said to be:
1. convex if tρ + (1 − t)ρ ⊆ ρ, (∀)t ∈ [0, 1];
2. balanced if λρ ⊆ ρ, (∀)λ ∈ K, |λ| ≤ 1;
3. absorbing if
∨
t>0
tρ = 1;
4. absolutely convex if it is both convex and balanced.
Proposition 45. [27] Let ρ be a fuzzy set in X. Then:
1. ρ is convex if and only if
ρ(tx + (1 − t)y) ≥ ρ(x) ∧ ρ(y), (∀)x, y ∈ X, (∀)t ∈ [0, 1];
2. ρ is balanced if and only if ρ(λx) ≥ ρ(x), (∀)x ∈ X, (∀)λ ∈ K, |λ| ≤ 1.
Definition 46. [29] A Katsaras fuzzy semi-norm on X is a fuzzy set ρ in X which is absolutely
convex and absorbing.
Proposition 47. [31] Let ρ be a Katsaras fuzzy semi-norm on X. Let
pα(x) := inf{t > 0 : ρ
(x
t
)
> α}, α ∈ (0, 1) .
Then P = {pα}α∈(0,1) is an ascending family of semi-norms on X.
Definition 48. [10] A fuzzy semi-norm ρ on X will be called Katsaras fuzzy norm if
ρ
(x
t
)
= 1, (∀)t > 0 ⇒ x = 0 .
Remark 49. a) It is easy to see that
[
ρ
(x
t
)
= 1, (∀)t > 0 ⇒ x = 0
]
⇔
[
inf
t>0
ρ
(x
t
)
< 1, for x 6= 0
]
.
b) The condition
[
ρ
(
x
t
)
= 1, (∀)t > 0 ⇒ x = 0
]
is much weaker than that one imposed by A.K.
Katsaras [29],
[
inf
t>0
ρ
(x
t
)
= 0, for x 6= 0
]
.
Proposition 50. [10] Let ρ be a Katsaras fuzzy semi-norm and
pα(x) := inf{t > 0 : ρ
(x
t
)
> α}, α ∈ (0, 1).
Then the family of semi-norms P = {pα}α∈(0,1) is sufficient if and only if ρ is a Katsaras fuzzy
norm.
Definition 51. [17] Let X be a vector space over R, let || · || : X → R∗(I) and let the mappings
L, R : [0, 1]×[0, 1] → [0, 1] be symmetric, nondecreasing in both arguments and satisfy L(0, 0) = 0
and R(1, 1) = 1. We write [||x||]α = [||x||
α
1 , ||x||
α
2 ], for x ∈ X, α ∈ (0, 1].
We suppose that (∀)x ∈ X, x 6= 0 there exists α0 ∈ (0, 1] independent of x such that for all
α ≤ α0 we have
(A) ||x||α2 < ∞ ,
(B) inf ||x||α1 > 0 .
The quadruple (X, || · ||, L, R) is called fuzzy normed linear space and || · || a fuzzy norm, if
The Fuzzification of Classical Structures: A General View 783
1. ||x|| = 0 if and only if x = 0;
2. ||rx|| = |r| · ||x||, (∀)x ∈ X, r ∈ R;
3. for all x, y ∈ X,
(a) whenever s ≤ ||x||11, t ≤ ||y||
1
1 and s + t ≤ ||x + y||
1
1,
||x + y||(s + t) ≥ L(||x||(s), ||y||(t)) ,
(b) whenever s ≥ ||x||11, t ≥ ||y||
1
1 and s + t ≥ ||x + y||
1
1,
||x + y||(s + t) ≤ R(||x||(s), ||y||(t)) .
Remark 52. C. Felbin [17] proved that, if L(x, y) = min{x, y} and R(x, y) = max{x, y}, then
the triangle inequality (3) in previous definition is equivalent to ||x + y|| ≤ ||x|| + ||y||. Further
|| · ||iα are crisp norms on X, for each α ∈ (0, 1] and i = 1, 2.
Remark 53. In paper [5], Felbin’s definition of fuzzy normed linear space is slightly modified in
the sense that:
1. the value of the fuzzy norm is taken to be a fuzzy real number in the sense of J.Z. Xiao
and X.H. Zhu [59];
2. the condition (A) and (B) of Felbin’s definition are relaxed by the condition
(A′) x 6= 0 ⇒ ||x||(t) = 0, (∀)t ≤ 0 .
Definition 54. [10] Let X be a vector space over a field K and ∗ be a continuous t-norm. A
fuzzy set N in X × [0,∞) is called a fuzzy norm on X if it satisfies:
(N1) N(x, 0) = 0, (∀)x ∈ X;
(N2) [N(x, t) = 1, (∀)t > 0] if and only if x = 0;
(N3) N(λx, t) = N
(
x, t
|λ|
)
, (∀)x ∈ X, (∀)t ≥ 0, (∀)λ ∈ K∗;
(N4) N(x + y, t + s) ≥ N(x, t) ∗ N(y, s), (∀)x, y ∈ X, (∀)t, s ≥ 0;
(N5) (∀)x ∈ X, N(x, ·) is left continuous and lim
t→∞
N(x, t) = 1.
The triple (X, N,∗) will be called fuzzy normed linear space (briefly FNL-space).
Remark 55. a) T. Bag and S.K. Samanta [2], [3] gave a similar definition for ∗ = ∧, but in order
to obtain some important results they assumed that the fuzzy norm satisfies also the following
conditions:
(N6) N(x, t) > 0, (∀)t > 0 ⇒ x = 0 ;
(N7) (∀)x 6= 0, N(x, ·) is a continuous function and strictly increasing on the subset {t : 0 <
N(x, t) < 1} of R.
The results obtained by T. Bag and S.K. Samanta can be found in these more general settings [10].
b) I. Goleţ [7], C. Alegre and S. Romaguera [1] gave also the same definition in the context
of real vector spaces.
Remark 56. N(x, ·) is nondecreasing, (∀)x ∈ X.
784 I. Dzitac
Theorem 8.1. [10] If (X, N,∗) is a FNL-space, then
M : X × X × [0,∞) → [0, 1], M(x, y, t) = N(x − y, t)
is a fuzzy metric on X, which is called the fuzzy metric induced by the fuzzy norm N. Moreover,
we have:
1. M is a translation-invariant fuzzy metric;
2. M(λx, λy, t) = M
(
x, y, t
|λ|
)
, (∀)x ∈ X, (∀)t ≥ 0, (∀)λ ∈ K∗.
Corollary 57. [10] Let (X, N,∗) be a FNL-space. For x ∈ X, r ∈ (0, 1), t > 0 we define the open
ball
B(x, r, t) := {y ∈ X : N(x − y, t) > 1 − r} .
Then
TN := {T ⊂ X : x ∈ T iff (∃)t > 0, r ∈ (0, 1) : B(x, r, t) ⊆ T}
is a topology on X.
Moreover, if the t-norm ∗ satisfies sup
x∈(0,1)
x ∗ x = 1, then (X,TN) is Hausdorff.
Theorem 8.2. [10] Let (X, N,∗) be a FNL-space. Then (X,TN) is a metrizable topological vector
space.
9 Conclusions
Lotfi A. Zadeh, born on February 4, 1921, is a famous mathematician, electrical engineer,
computer scientist, and Professor Emeritus at the University of California, Berkeley, United State
of America. He is father of fuzzy sets, fuzzy logic and computing with words. His pioneering
paper, entitled "Fuzzy Sets" (1965, [61]), is cited over 58,540 time in many prestigious journals,
and all his papers are cited over 151,300 time.
Some scientists, especially philosophers and mathematicians, had attempted to formalize the
process of logical deduction. Their work culminated in the invention of the programmable digital
computer, a machine based on the abstract essence of mathematical reasoning. This machine
and the ideas behind it inspired a handful of scientists to begin seriously discussing the possibility
of building an artificial brain.
In this survey paper we mentioned some fuzzy mathematical structures as fuzzy real line,
fuzzy topological spaces, fuzzy metric spaces, fuzzy topological vector spaces, fuzzy normed linear
spaces and fuzzy F-space.
Acknowledgments
As an editor of this Special Issue on Fuzzy Sets and Applications, dedicated to celebration
of the 50th anniversary of Fuzzy Sets, I should like to express my deep appreciation to the
distinguished authors from Belgium, Canada, China, Greece, Lithuania, Pakistan, Romania,
Spain, Taiwan, and USA, for their valuable contributions (in order of appearance in this issue):
R.R. Yager, S. Ashraf, A. Rehman, E.E. Kerre, O. Bologa, R.E. Breaz, S.G. Racz, Z.-B. Du,
T.-C. Lin, T.-B. Zhao, V. Kreinovich, C. Stylios, S. Nădăban, R.-E. Precup, M.L. Tomescu, E.M.
Petriu, X. Tang, L. Shu, H.-N.L. Teodorescu, Z. Turskis, E.K. Zavadskas, J. Antucheviciene, N.
Kosareva, H. Xu, R. Vilanova, H.D. Wang, S.C. Guo, S. M. Hosseini Bamakan, Y. Shi, T. Wang,
G. Zhang, M.J. Pérez-Jiménez, and F.G. Filip.
The Fuzzification of Classical Structures: A General View 785
Also many thanks to Sorin Nădăban (Romania) and Bogdana Stanojevic (Serbia) for their
valuable comments that help improve the manuscript.
It is my great pleasure to thanks Prof. Lotfi A. Zadeh for his friendly collaborations with
me in 2008 (Fig.1, [65], [65]), in 2011 [64], and for his acceptation and helpful collaboration in
editing of this special issue in 2015. Celebrating the 50th anniversary of your pioneering paper
"Fuzzy Sets", and your 94th birthday, I wish you, dear professor Zadeh, a good health, long life,
and new interesting achievements.
(a) Lotfi A. Zadeh & Ioan Dzitac at ICCCC 2008 (b) Ed. by L.A. Zadeh, D. Tufis, F.G. Filip, I. Dzitac
Figure 1: Meeting with Professor Lotfi A. Zadeh (Agora University of Oradea, Romania, 2008)
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