THE IMPACT OF FUZZY SET THEORY ON

Appl. Comput. Math., V.10, N.1, Special Issue, 2011, pp.20-34

THE IMPACT OF FUZZY SET THEORY ON CONTEMPORARY MATHEMATICS SURVEY ETIENNE E. KERRE1 Abstract. In this paper we first outline the shortcomings of classical binary logic and Cantor’s set theory in order to handle imprecise and uncertain information. Next we briefly introduce the basic notions of Zadeh’s fuzzy set theory among them: definition of a fuzzy set, operations on fuzzy sets, the concept of a linguistic variable, the concept of a fuzzy number and a fuzzy relation. The major part consists of a sketch of the evolution of the mathematics of fuzziness, mostly illustrated with examples from my research group during the past 35 years. In this evolution I see three overlapping stages. In the first stage taking place during the seventies only straightforward fuzzifications of classical domains such as general topology, theory of groups, relational calculus, . . . have been introduced and investigated w.r.t. the main deviations from their binary originals. The second stage is characterized by an explosion of the possible fuzzifications of the classical structures which has lead to a deep study of the alternatives as well as to the enrichment of the structures due to the non-equivalence of the different fuzzifications. Finally some of the current topics of research in the mathematics of fuzziness are highlighted. Nowadays fuzzy research concerns standardization, axiomatization, extensions to lattice-valued fuzzy sets, critical comparison of the different so-called soft computing models that have been launched during the past three decennia for the representation and processing of incomplete information. Keywords: Fuzzy Set Theory, Fuzzy Mathematics, Survey, Historical Overview. AMS subject classification: 94D05.

1. Introduction Most of the information we have to tackle in daily life is pervaded with imprecision and uncertainty, two facets of the incompleteness of information. Till the mid sixties there were no mathematical models available in order to treat imprecise information. What happened for so many centuries is the following. Consider a universe of patients in some hospital. Several (crisp or black-or-white) properties can be considered concerning each of these patients: being married, being divorced, being epileptic, systolic blood pressure of 12, being 1.70m high, weight of 74kg, . . . For each patient some of these properties hold (leading to a true statement) or not (leading to a false statement). Using Aristotelian or binary logic several properties may be combined into more complex ones by means of the well-known connectives such as negation (NOT), disjunction (OR) and conjunction (AND). The truth value of these compositions is determined by means of truth tables. For example the conjunction “P AND Q” of two propositions P and Q is only true when both propositions are true. It is well known that the class of propositions together with the unary operation of negation and the binary operations of disjunction and conjunction constitutes the very strong structure of a Boolean algebra, so majestically axiomatically characterized by Huntington in 1904 (for more details see [75]). We want to mention here that classical logic satisfies 1

Fuzziness and Uncertainty Modelling, Department of Applied Mathematics and Computer Science, Ghent University, Krijgslaan 281 (S9), B–9000 Gent, Belgium e-mail: [email protected], URL: http://www.fuzzy.ugent.be Manuscript received 12 December, 2010. 20

ETIENNE E. KERRE: THE IMPACT OF FUZZY SET THEORY ...

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two important laws (also called tautologies, i.e., statements that are always true): the law of the excluded middle stating that “P OR NOT P ” is always true and the law of contradiction stating that “NOT (P AND NOT P )” is always true. At the end of the 19th century G. Cantor introduced the famous set theory that has been widely accepted as the language of science in general and of mathematics in particular. For a given universe of discourse X (for example patients, students, cars, diseases, companies, . . . ) each crisp property leads to a crisp partition of the universe into the (sub-)set A of elements of X that satisfy this property and the remaining (sub-)set co A of elements of X that do not satisfy this property. Corresponding to the logical connectives Cantor introduced on (sub-) sets the operations of complementation, union and intersection. For example consider the sets A = {x | P } and B = {x | Q} then the complement of A, the union and intersection of A and B are defined as co A = {x | NOT P }, A ∪ B = {x | P OR Q}, A ∩ B = {x | P AND Q}. As a direct consequence of the Boolean structure of propositional logic the class of all subsets of a universe X, i.e., the so-called powerclass P(X) of X, together with the operations of complementation, union and intersection also constitutes a Boolean algebra. Now let’s go back to our universe of patients and consider properties such as: being tall, being more-or-less corpulent, being middle-aged, . . . In order to represent such qualitative, imprecise predicates one has to specify boundaries. For example some patient is considered tall as soon as his length exceeds 1.85m. As a consequence of this sharp boundary the patient that measures 1.84m will be classified as being not tall, while practically there is almost no difference between 1.84m and 1.85m. In this respect we have to cite the so-called Poincaré paradox [35]. Poincaré observed that a human cannot determine by hand a difference between a package of 100g and one of 101g, so for a human observer both packages have the same weight i.e., they are equal with respect to weight. Similarly there is no difference between 101g and 102g and hence applying the transitivity property of the equality relation leads to the conclusion that there is no difference between 100g and 102g. It is easy to see that continuing this reasoning from classical logic induces the conclusion that all packages have the same weight. . . Similar paradoxes were already known by the old Greeks during the 5th century B.C.: Sorites and Falakros paradox. Another example of making imprecise concepts precise is the following definition of a red object: a red object is one indistinguishable in hue from a uniformly reflecting surface with monochromatic light of wavelength between 580.27nm and 702.35nm. . . So putting crisp boundaries in order to define imprecise concepts leads to unworkable, artificial situations. My favorite example comes from B. Gaines [63]. The sentence “List the young salesmen who have a good selling record for household goods in the north of England” is despite the imprecise terms (young, good, household goods, north) perfectly comprehensible. However in order to make this sentence comprehensive to a classical retrieval system we need a transformation as follows: “List salesman under 25 years old who have sold more than more £20.000 of goods in the categories . . . to shops in the regions . . . ” As a consequence of this precisation a classical retrieval system will not keep the guy of 26 years old with a very good selling record nor the one of 19 years with a selling record of £19.000. Before moving into the basics of Zadeh’s fuzzy sets I would like to stress the consequences of binary logic and classical set theory on mathematics. As soon as the classifier { | } has been introduced – intuitively or axiomatically – the basic concept of a relation between two sets can be defined. Then – too soon to my opinion – the concept of a relation has been narrowed into a functional one or shortly a function. In particular a function with the natural numbers as domain is known as a sequence. The class of sequences is again divided into two disjoint subsets; the convergent and the non-convergent sequences. This bivalent partitioning is very

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APPL. COMPUT. MATH., V.10, N.1, SPECIAL ISSUE, 2011

rough as illustrated by the following example. Consider the sequence (1, −2, 3, −4, 5, . . .), it is clearly a non-convergent one. Similarly the sequence (0.01, −0.01, 0.01, −0.01, 0.01, . . .) is also not convergent. So both sequences belong to the same class of non-convergent real sequences while there is a huge difference between them: the first one is totally hopeless with respect to being convergent while the second one is “almost” convergent to zero. So if convergence could be introduced as a gradual notion (with more than two degrees) then the second sequence certainly would get a high degree. As illustrated so far there is a tremendous need for mathematical models to represent and process imprecise and uncertain information. Till the mid sixties only probability theory and error calculus were partly able to satisfy the need for a special kind of uncertainty namely randomness. As stated by Zadeh: probability theory is insufficiently expressive to serve as the language of uncertainty. It has no facilities to describe fuzzy predicates such as small, young, much larger than, . . . , nor fuzzy quantifiers such as most, many, a few, . . . , nor fuzzy probabilities such as likely, not very likely, . . . , nor linguistic modifiers such as very, more-or-less, . . . We had to wait till 1965 when L. Zadeh launched his seminal paper “Fuzzy sets” [117]. In this paper the concept of a fuzzy set allows to have besides membership and non-membership, intermediate or partial degrees of membership. So instead of black-or-white decisions a gradual transition from membership to non-membership has been introduced. Let’s have a closer look to the basic concepts of this remarkable, innovative theory that counted in October 2010 already 71,521 papers in the INSPEC database with fuzzy in the title and 18,863 papers in the MATH.SCI.NET database! 2. Basic notions from fuzzy set theory Let X be an arbitrary non-empty set, the so-called universe of discourse. A mapping A from X into the unit interval [0, 1] is called a fuzzy set on X. The value A(x) of A in the element x of X is called the degree of membership of x in the fuzzy set A; A(x) = 1 means full membership and A(x) = 0 means non-membership. The choice of the membership function is context dependent and within the same context dependent on the observer. Although the knowledge of an exact value for the membership degree is not really needed it is sometimes useful to have some general shape for the membership function available. Such general shape functions are dependent on some parameters that can be adapted to the context as well as to the observer. Well-known examples of such typical membership functions are the S-membership function and the π-membership function [76]. For an excellent overview of various interpretations of the membership function as well as many ways to obtain such membership functions we refer to an extensive chapter of T. Bilgi¸c and B. T¨ urk¸sen in the Handbooks of Fuzzy Sets [9]. The choice of the unit interval for the degrees of membership implies that every two objects should be comparable w.r.t. an imprecise predicate. In order to tackle situations where also incomparability appears, Goguen [66] extended Zadeh’s fuzzy set theory into lattice valued fuzzy set theory or shortly L-fuzzy set theory where usually L denotes a complete lattice. So an L-fuzzy set in a universe X is a mapping from X into the complete lattice L. The class of L-fuzzy sets can also be endowed with union and intersection operations using the supremum and infimum in the lattice L. The following notions will be useful for a fuzzy set A in X: • the support of A: supp A = {x | A(x) > 0}, • the kernel of A: ker A = {x | A(x) = 1}, • the weak α-level of A: Aα = {x | A(x) ≥ α}, • the strong α-level of A: Aα¯ = {x | A(x) > α}, • the height of A: hgt A = sup{A(x) | x ∈ X}, • the plinth of A: plt A = inf{A(x) | x ∈ X}. The class of all fuzzy sets in a universe X will be denoted as F(X).


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The set-theoretic operations have been extended to fuzzy sets as follows. Let A and B be two fuzzy sets in the universe X. Then we define: • the complement co A of A as the fuzzy set in X with degree of membership in x: co A(x) = 1 − A(x), • the union A ∪ B of A and B as the fuzzy set in X with degree of membership in x: A ∪ B(x) = max(A(x), B(x)), • the intersection A ∩ B of A and B as the fuzzy set in X with degree of membership in x: A ∩ B(x) = min(A(x), B(x)). It is well known that the structure (F(X), ∪, ∩, co) constitutes a soft or Morgan algebra, i.e., a bounded, completely distributive lattice satisfying the additional properties: • co ∅ = X and co X = ∅, • co(A ∪ B) = co A ∩ co B and co(A ∩ B) = co A ∪ co B, • co(co A) = A. The most important deviations of the Morgan algebra (F(X), ∪, ∩, co) of fuzzy sets in X from the Boolean algebra (P(X), ∪, ∩, co) of crisp subsets of X are the following: (i) Fuzzy set theory violates the law of the excluded middle. Hence a fuzzy set and its complement do not necessary fill up the whole universe. Only a weakened law of the excluded middle holds, stating that a fuzzy set and its complement fill up at least half of the universe, i.e. µ ¶ 1 (∀x ∈ X) A ∪ co A(x) ≥ . 2 (ii) Fuzzy set theory violates the law of contradiction. Hence there may be some overlap between a fuzzy set and its complement. Only a weakened law of contradiction holds, stating that the overlap between a fuzzy set and its complement being at most half of the universe, i.e. µ ¶ 1 (∀x ∈ X) A ∩ co A(x) ≤ . 2 (iii) Classically for crisp subsets A and B of X, from co A ⊆ B we can infer A ∪ B = X. This property no longer holds for fuzzy sets in X. (iv) Similarly for crisp subsets A and B of X, from B ⊆ co A we can infer A ∩ B = ∅. This property no longer holds for fuzzy sets in X. This deviation has great impact on the fuzzification of notions involving disjoint sets. It should be stressed that maximum and minimum are not the only possible operations to generalize classical union and intersection. A lot of theoretical and practical research has been performed on the so-called confluence of degrees of membership [8, 123]. In fact every triangular norm, respectively triangular conorm [105] can be used to model intersection, respectively union. A triangular norm T (respectively conorm S) is a [0, 1]2 −[0, 1] mapping satisfying commutativity, associativity, non-decreasingness and the boundary condition T (x, 1) = x (respectively S(x, 0) = x) for every x ∈ [0, 1]. Popular choices for T in fuzzy set theory are: algebraic product (a · b) and bold intersection (also known as Lukasiewicz t-norm) defined by W (a, b) = max(0, a + b − 1). The corresponding dual conorms S are the probabilistic sum (a + b − ab) and the bounded sum (min(1, a + b)). From practical point view maybe the most important concept in fuzzy set theory is the concept of a linguistic variable, i.e., a variable that takes linguistic instead of numerical values. For example the unemployment rate in some country is considered as a numerical value if it takes values such as 10%, 12.5%, . . . and as a linguistic variable if it takes values such as low, very high, more-or-less non-existing, . . . Zadeh [119] observed that the set of values for a linguistic variable assumes a more-or-less fixed structure consisting of: • an atomic term (e.g. low), • the antonym of the atomic term (e.g. high),

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• logical combinations using NOT, AND, OR (e.g. NOT low, NOT low AND NOT high), • linguistic hedges or modifiers (e.g. very, more-or-less). To represent the atomic term and its antonym the parameterized membership functions may be used. The logical combinations may be modeled by means of the triangular norms and conorms. For the representation of the linguistic hedges several specific operators are available: concentration, dilatation, Zadeh’s contrast intensification, Gupta and Ragade’s contrast intensification, support fuzzification and the shifting operations [75]. From theoretical point of view one of the most important contributions of Zadeh is undoubtedly the extension principle [119], allowing to extend to fuzzy set theory almost every mathematical concept and structure based on binary logic and set theory. For a detailed overview of this principle and its extensions we refer to [71]. Since the introduction by Mizumoto and Tanaka [89] in 1979 of the concept of a fuzzy number as a fuzzy set on the real line satisfying some convexity and continuity properties a vast amount of papers has been published on this topic [75]. Due to the extension principle all the unary (opposite, multiplication by a real number, translation by a real number, exponential function, . . . ) and all the binary operations (addition, multiplication, subtraction, division, . . . ) have been extended to fuzzy numbers. For a more-or-less recent survey of the state of the art about fuzzy numbers and especially fuzzy intervals (i.e., fuzzy sets in the real line whose weak α-level sets are intervals) we refer to [59]. As already mentioned before the concept of a relation is a fundamental one and deserves much more attention than its special functional instance, because most of the relationships we encounter are not functional. A relation between two sets X and Y is defined as a (crisp) subset of the Cartesian product X × Y . Hence all the set-theoretic operations can be applied to relations. These operations have been very useful for the development of relational databases. But of course the most important operation between relations is the product or composition of relations. Important information can be obtained from a single relation by using the concepts of direct and inverse image. Just as the problem of deciding whether some object satisfies a given property i.e., belongs to a given set of objects, has to be considered as a matter of degree rather than a yes-or-no decision, it is more realistic to agree that instead of only considering ordered pairs that are related or not we have to admit partial degrees of relationship. More formally a fuzzy relation R from X to Y is defined as a fuzzy set in X × Y , where for every ordered pair (x, y) ∈ X × Y , R(x, y) is interpreted as the strength of the existing R-link between x and y. All the set-theoretic operations, the composition, the direct and inverse images have been extended to fuzzy relations [75]. 3. On the evolution of the mathematics of fuzziness First of all I would like to emphasize that I prefer the term “mathematics of fuzziness” instead of “fuzzy mathematics” because there is nothing fuzzy or blurry in this kind of mathematics. When I started my research in fuzzy set theory in 1976 only a few hundreds of papers were published and hence very manageable to survey. Nowadays there are more than 71,000 papers with fuzzy in the title in the INSPEC database and almost 19,000 papers in the MathSciNet database, and hence it is completely impossible to write a survey paper. So shall I stop here or shall I try to give a flavor of what Zadeh’s 1965 paper has signified for the evolution of the contemporary mathematics? Let’s go for the latter option! After a short dissemination period for Zadeh’s brilliant concept of fuzzy sets mathematicians became aware of the enormous possibilities of this theory for extending the existing mathematical apparatus, especially with regard to the applications, since this concept embraces the elasticity of natural language and human’s qualitative summarization capabilities. Probably the first domain of mathematics that underwent the coloring process was general topology. Indeed already in 1969 C. L. Chang published his seminal paper entitled “Fuzzy topological spaces” [13]. A fuzzy topology on a universe X is a subclass τ of the class of fuzzy


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sets on X that contains (i) the empty set (identified with the constant mapping of X onto 0) and the universe X ( identified with the constant mapping of X onto 1), (ii) contains the (Zadeh) intersection of every two elements of τ and (iii) contains the (Zadeh) union of each arbitrary family of elements of τ . In 1976 R. Lowen [86] strengthened this Chang definition by requiring that not only the constant mappings on 0 and 1 should belong to τ but every constant mapping from X onto k for every k ∈ [0, 1] should belong to τ . Such a fuzzy topology is called a stratified fuzzy topology. A fuzzy topology is a typical example of a straightforward fuzzification of a classical structure: replace the class of crisp subsets by the class of fuzzy sets and the settheoretic operations by the fuzzy ones. That’s what mostly happened during the first 15 years following the birth of fuzzy sets. In this way straightforward generalizations of topology related notions such as closed fuzzy set, interior, closure and neighborhood were introduced. Once the fuzzy concept was introduced the main topic of research consisted of the determination of those classical properties that remain valid or not. I took part in this process by looking at the characterization of a fuzzy topology by means of preassigned operations. We could easily prove that a fuzzy topology could be characterized by means of the closed fuzzy sets, by means of an interior operator and by means of a closure operator. We however initially failed in proving or disproving a characterization in terms of neighborhood systems for each fuzzy singleton. The main reason for this failure turned out to be the fact that a fuzzy singleton could “belong” to an arbitrary union of fuzzy sets without belonging to one of them [80]. Several notions of a fuzzy neighborhood have been introduced and extensively studied later on. For a detailed overview of the different notions, their interrelationships and the ultimate solution of the characterization problem we refer to [79, 80, 81, 82]. The fuzzification of algebraic structures such as groups and ideals of groups has been initiated by A. Rosenfeld [100] in 1971. Let (X, ·) be a groupoid and A a fuzzy set in X. The structure (A, ·) is called a fuzzy groupoid iff for each (x, y) belonging to X the inequality A(x · y) ≥ min(A(x), A(y)) holds. Again it is easy to see that this notion extends the crisp notion of a subgroupoid. Moreover Rosenfeld established for the first time an interesting relationship between a fuzzy notion and its underlying crisp notion via the concept of the α-level sets: (A, ·) is a fuzzy subgroupoid of (X, ·) iff (Aα , ·) is a subgroupoid for each α belonging to ]0, 1]. In other fuzzified structures this characterization holds for the weak as well as for the strong α-levels (for example the characterization of a similarity relation in terms of the underlying equivalence relation) and sometimes it holds for only one kind of α-levels. So everytime when a crisp notion has been fuzzified the question about such a characterization arises and also vice versa: under which conditions can a fuzzy structure be reconstructed from the knowledge of its α-levels? A lot of research in this respect has been done for a fuzzy topology by Lowen and Wuyts. The concept of a fuzzy module has been introduced by Negoita and Ralescu [94] in the first monograph textbook on fuzzy sets that appeared already in 1975. Katsaras [70] was the first to introduce and investigate fuzzy vector spaces in 1977. For an excellent overview of what has been produced on fuzzy algebraic structures, especially fuzzy group theory we refer to the books of J. Mordeson et al. [90, 91]. In [85] Kuijken, Van Maldeghem and Kerre introduced the concept of a fuzzy projective geometry as a natural link between fuzzy vector spaces and fuzzy groups. Other examples of straightforward fuzzifications of classical structures are: the calculus of fuzzy relations, fuzzy metric spaces, fuzzy measure theory and fuzzy integrals, the latter ones introduced in the famous Ph. D. thesis of M. Sugeno [110]. In the eighties a new stage in the development of the mathematics of fuzziness started. The introduction of the notions of triangular norm and co-norm from probabilistic metric spaces [105] has lead to an explosion of the possible choices in the process of fuzzification. In this way concepts such as T -similarity relation, T -fuzzy groups, T -partitions have been introduced. These generalizations have lead to a deep study of the alternatives for maximum and minimum on a logical level (negation, disjunction, conjunction, implication) as well as on a fuzzy-set-theoretic

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level (complementation, union, intersection, inclusion). Another characteristic of this second stage is the diversification that was due to the non-equivalence of the different generalizations of basic concepts and structures. Let’s give one example: the extension of the concept of normality to fuzzy topological spaces. A crisp topological space (X, T ) is called normal iff: (∀(F1 , F2 ) ∈ (T 0 )2 )(F1 ∩ F2 = ∅ =⇒ (∃(O1 , O2 ) ∈ T 2 )(O1 ∩ O2 = ∅, F1 ⊆ O1 and F2 ⊆ O2 )), where T 0 denotes the class of closed sets, i.e., T 0 = {F | co F ∈ T }. It is well known that Urysohn proved the following characterization of normality: (X, T ) is normal ⇐⇒ (∀O ∈ T )(∀F, F ∈ T 0 and F ⊆ O) (∃V ∈ P(X))(F ⊆ int V and cl V ⊆ O), where int V (cl V ) denotes the interior (closure) of V . Now let (X, τ ) be a fuzzy topological space in the sense of Chang. In 1975 B. Hutton introduced the following definition of normality for a fuzzy topological space: (X, τ ) is normal ⇐⇒ (∀O ∈ τ )(∀F, F ∈ τ 0 and F ⊆ O) (∃V ∈ F(X))(F ⊆ int V and cl V ⊆ O), τ0

where similarly denotes the class of closed fuzzy sets, i.e., τ 0 = {F | co F ∈ τ }, i.e., a straightforward fuzzification of Urysohn’s characterization. The following question came into my mind: why did Hutton take Urysohn’s form to fuzzify normality and not the original definition? The answer has been given in [72]. To fuzzify the concept of normality we need a suitable fuzzification of disjointness of sets. As said before the following equivalence holds for crisp sets: F1 ∩ F2 = ∅ ⇐⇒ F1 ⊆ co F2 . For fuzzy sets however only the implication F1 ∩ F2 = ∅ =⇒ F1 ⊆ co F2 holds. Based on this observation we introduced the following concepts in a fuzzy topological space (X, τ ): (X, τ ) is normal ⇐⇒ (∀(F1 , F2 ) ∈ (τ 0 )2 ) (F1 ⊆ co F2 =⇒ (∃(O1 , O2 ) ∈ τ 2 )(O1 ⊆ co O2 , F1 ⊆ O1 and F2 ⊆ O2 )), (X, τ ) is weakly normal ⇐⇒ (∀(F1 , F2 ) ∈ (τ 0 )2 ) (F1 ∩ F2 = ∅ =⇒ (∃(O1 , O2 ) ∈ τ 2 )(O1 ⊆ co O2 , F1 ⊆ O1 and F2 ⊆ O2 )), (X, τ ) is completely normal ⇐⇒ (∀(A1 , A2 ) ∈ (F(X))2 )(A1 ⊆ co(cl A2 ) and A2 ⊆ co(cl A1 ) =⇒ (∃(O1 , O2 ) ∈ τ 2 )(O1 ⊆ co O2 , A1 ⊆ O1 and A2 ⊆ O2 )). It is easily seen that for crisp topological spaces both concepts of normality and weakly normality coincide with the classical concept given above. For a fuzzy topological space however we could only prove the following properties. P.1 (X, τ ) is completely normal =⇒ (X, τ ) is normal, P.2 (X, τ ) is normal =⇒ (X, τ ) is weakly normal, P.3 (X, τ ) is normal ⇐⇒ (X, τ ) is normal in the sense of Hutton, P.4 (X, τ ) is completely normal =⇒ every subspace of (X, τ ) is normal.


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We could prove that the converse of P.2 is not valid, since the fuzzy Sierpinski space [73] is a counterexample. From P.3 it can be seen that our definition of normality i.e., interpreting the disjointness of F1 and F2 as F1 ⊆ co F2 , leads to the fuzzy analogue of Urysohn’s theorem. We however were not able to prove the converse of P.4, i.e., the fuzzy extension of Tietze’s characterization theorem of complete normality. Such deviations occur frequently and are quite normal because fuzzy set theory constitutes a generalization of (crisp) set theory. Other examples of the explosion in the alternatives in the fuzzy generalizations can be found in the ranking methods for fuzzy quantities, i.e., how to compare imprecise quantities such as “about 5”, “approximately between 3 and 7”, . . . For an introduction of these ranking methods as well as a critical mutual comparison we refer to [11, 69, 116]. Now let’s finally turn to highlight some of the current topics of research in mathematics of fuzziness. Starting in the nineties the following are among the main themes of fuzzy research: standardization, axiomatization, L-fuzzification and a critical comparison of the fuzzy model to the many other models that were build during the last two decennia for the representation and processing of imprecise and uncertain information. The purpose of the standardization process is the motivated selection of the most suitable definition among the many possibilities that were introduced during the explosion in the eighties. We have to agree upon the most suitable fuzzified version of a point or an atom in set theory: fuzzy point or fuzzy singleton and which corresponding membership relation (subset relation, quasi-coincidence relation or belongingness relation). Which concept of neighborhood will be selected (Ludescher, Kerre, Warren, Pu, Mashhour)? Which triangular norm and co-norm will be used to represent the extension of the logical or set-theoretic operators? Many papers appeared that are entitled: fuzzy this or that revisited. So standardization is needed: the selection of the most suitable definition is very important and should be based upon a good balancing of the pro’s and contra’s. Then once the choice has been made, every deviation from the standard should be clearly specified and should not use the same name. As an example let’s take the concept of continuity of a mapping between two topological spaces. Firstly let f be a mapping between two crisp topological spaces (X1 , T1 ) and (X2 , T2 ). The following equivalent formulations of the continuity of f are well known: f is continuous ⇐⇒ (∀O2 ∈ T2 )(f −1 (O2 ) ∈ T1 ) ⇐⇒ (∀F2 ∈ T20 )(f −1 (F2 ) ∈ T10 ) ⇐⇒ (∀A1 ∈ P(X1 ))(f (cl(A1 )) ⊆ cl f (A1 )) ⇐⇒ (∀A2 ∈ P(X2 ))(f −1 (int(A2 )) ⊆ int f −1 (A2 )) ... Now let f be a mapping between two fuzzy topological spaces (X1 , τ1 ) and (X2 , τ2 ). In [13] Chang introduced the following definition for the fuzzy continuity of f : f is fuzzy continuous ⇐⇒ (∀O2 ∈ τ2 )(f −1 (O2 ) ∈ τ1 ), where the inverse image of the open fuzzy set O2 under the mapping f is defined f −1 (O2 )(x) = O2 (f (x)) for every x ∈ X1 . During the eighties already 26 different forms of fuzzy continuity have been introduced: fuzzy weak continuous, fuzzy almost continuous, fuzzy semicontinuous, fuzzy weak semicontinuous, fuzzy semicontinuous, fuzzy almost semicontinuous, fuzzy δ-continuous, fuzzy semi irresolute continuous, fuzzy strongly irresolute continuous, fuzzy θcontinuous, fuzzy strongly θ-continuous, fuzzy almost strongly θ-continuous, fuzzy super continuous, fuzzy weak θ-continuous, fuzzy weak precontinuous, fuzzy weak λ-continuous, fuzzy semi strongly θ-continuous, fuzzy (θ, S)-continuous, fuzzy quasi irresolute continuous, fuzzy λirresolute continuous and fuzzy semi weak continuous. This example illustrates once more the diversification of notions, that in the crisp case coincide. In 1991 we introduced [69, 78] the

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notion of an operation on a fuzzy topological space by which we could obtain all the previous forms of fuzzy continuity as special instances. Some examples of axiomatic systems for fuzzy concepts that have been developed during the last two decennia are the following. • Axiomatic system for defuzzification operators. In fuzzy control and in expert systems usually a fuzzy set is obtained as conclusion of applying approximate reasoning techniques such as the compositional rule of inference. This fuzzy set has to be defuzzified into a single quantity. A lot of defuzzification techniques has been introduced into the literature. In order to make a critical comparison between these defuzzification techniques Van Leekwijck and Kerre [112] established an axiomatic system of desirable properties for such a defuzzification operator and they classified the existing techniques w.r.t. these axioms. • Axiomatic system for fuzzy implication operators. In the fuzzy literature three main classes of fuzzy implication operators have been introduced: S-(strong) implications, R-(residual) implications and QL-(quantum logic) implications. For an extensive analytic as well as an algebraic study of these three classes we refer to the excellent monograph of Baczy´ nski and Jayaram [5]. Already in 1987 Smets and Magrez introduced an axiomatic system for a fuzzy implication [109]. Afterwards several desirable properties have been added leading to 13 different axioms to be potentially satisfied for a fuzzy implication. A deep study on the interrelationships (dependencies and independencies) between these axioms can be found in [107, 106]. For an extension of fuzzy implications to intuitionistic fuzzy sets we refer to [19, 55, 53]. • Axiomatic system for fuzzy inclusion measures. Such an axiomatic system has been initiated by Sinha and Dougherty [108] and independently by Kitainik [84] and further refined by Cornelis et al. [25]. Another important domain of research in the mathematics of fuzziness that is also launched by Zadeh [118] is fuzzy relational calculus. As already said before the concept of a relation is a fundamental one from theoretical as well as practical point of view. Classical relations are nowadays used in several domains such as relational databases, information retrieval, medical diagnosis, spatial and temporal information, image processing. For an up-to-date book of reference on relations where also in Chapter 14 some connections to the fuzzy extension have been mentioned, among them Sugeno’s fuzzy integral and Dempster-Shafer’s theory of evidence, we refer to [102]. Especially because of the numerous practical applications mathematicians have rediscovered the power of relational calculus. This power has been increasingly enlarged due to the coloring process of fuzzy set theory. First of all I want to mention here the important contributions of Bandler and Kohout [6] on crisp as well as fuzzy relational calculus by introducing three new ways of composing relations based on the concepts of after- and foresets. Let R be a relation from X to Y and S a relation from Y to Z. Then the classical round composition R ◦ S (read as: R before S) of R and S, as well as its refinements: the subcomposition R / S, the supercomposition R . S and the square composition R ¤ S are defined by: (x, z) ∈ R ◦ S ⇐⇒ xR ∩ Sz 6= ∅, (x, z) ∈ R / S ⇐⇒ xR ⊆ Sz, (x, z) ∈ R . S ⇐⇒ xR ⊇ Sz, (x, z) ∈ R ¤ S ⇐⇒ xR = Sz, where xR denotes the R-afterset of x defined by xR = {y | (x, y) ∈ R} and Sz is the S-foreset of z defined by: Sz = {y | (y, z) ∈ S}. These compositions have been extended to the fuzzy framework using fuzzy implication operators. Later on in a series of papers we have introduced new images in the classical as well as in the fuzzy case [26, 27, 28, 74]. Since the pioneering work of Bandler and Kohout a huge number of theoretical as well as practical issues in fuzzy relational


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calculus have been successfully tackled: preference modeling [29, 30, 31, 111], land evaluation [67], relational databases [14], ordering procedures [115], linguistics [36], image processing [92], topology [10, 34], information retrieval [83], simulation [87], temporal information [103] and spatial information [104]. Fuzzy set theory has not been the only model that has been introduced to treat imprecise and uncertain information. During the last three decennia several new models have been developed to mathematically tackle incomplete information. Some models are extensions of fuzzy set theory and others use a different path. The following theories were created after Zadeh’s paper on fuzzy sets: • • • • • • • • • • • • • • • • •

L-fuzzy set theory, introduced by J. Goguen in 1967 [66], Flou set theory, introduced by Y. Gentilhomme in 1968 [65], L-flou set theory, introduced by C. Negoita and D. Ralescu in 1975 [94], Type-2 fuzzy set theory, introduced by L. Zadeh in 1975 [119], Interval-valued fuzzy set theory, introduced by R. Sambuc in 1975 [101], Probabilistic set theory, introduced by K. Hirota in 1981 [68], Rough set theory, introduced by Z. Pawlak in 1982 [95], Intuitionistic fuzzy set theory, introduced by K. Atanassov in 1983 [2, 3], Twofold fuzzy set theory, introduced by D. Dubois and H. Prade in 1987 [60], Grey set theory, introduced by J. L. Deng in 1989 [43], Fuzzy rough set theory, introduced by D. Dubois and H. Prade in 1990 [61], Rough fuzzy set theory, introduced by D. Dubois and H. Prade in 1990 [61], Theory of imprecise probabilities, introduced by P. Walley in 1991 [114], Soft set theory, introduced by K. Basu, R. Deb and P. K. Pattanaik in 1992 [7], Toll set theory, introduced by D. Dubois and H. Prade in 1993 [62], Vague set theory, introduced by W. L. Gau and D. J. Buehrer in 1993 [64], Bipolar fuzzy set theory, introduced by W.-R. Zhang in 1994 [121].

Very soon it became clear that not all these models were completely independent from each other. In Figure 3 that resulted from results we obtained in [47, 51, 57, 77] some of the relationships are shown. In this diagram an arrow from theory A to theory B means that theory B is a generalization of theory A. Type-2 Fuzzy Sets

Vague Sets

Crisp Sets

Fuzzy Sets

Interval-Valued Fuzzy Sets

Intuitionistic Fuzzy Sets

Probabilistic Sets

Interval-Valued Intuitionistic Fuzzy Sets

Grey Sets

L-Fuzzy Sets

Intuitionistic L-Fuzzy Sets

Soft Sets

Figure 1. The links between different models of fuzziness.

Among these new models the most developed ones are Pawlak’s rough set theory and Atanassov’s intuitionistic fuzzy set theory. For more detailed information about recent progress about rough sets in our team we refer to [16, 32, 33, 37, 38, 93, 96, 97, 98, 99]. For more information about recent developments in intuitionistic fuzzy set theory or equivalently interval-valued fuzzy set theory we refer to [1, 4, 15, 17, 18, 20, 21, 22, 23, 24, 44, 45, 46, 48, 49, 50, 52, 53, 54, 55, 56, 58, 88]. Moreover an extensive survey of interval-valued fuzzy sets can be found in the paper of H. Bustince et al. [12]. A good survey of the algebra of logical operations on type-2 fuzzy sets is

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given by C. Walker and E. Walker [113]. For more information about bipolar fuzzy sets we refer to the recent monograph of W.-R. Zhang [122]. Closely related to the theory of fuzzy sets and mostly used for the representation of a specific kind of uncertainty, different from randomness is possibility theory, also initiated by L. Zadeh in 1978 [120]. In [39] G. de Cooman has shown an interesting formal analogy between possibility theory (by using an extension of Sugeno’s fuzzy integral) and probability theory (based on Lebesgue integral). In a series of papers [40, 41, 42] G. de Cooman has given a thorough treatment on the fundamentals of possibility theory. 4. Conclusion In this paper I have tried to give a flavor of the impact of Zadeh’s creation of fuzzy set theory on the development of contemporary mathematics and I sincerely hope it may motivate young researchers, especially mathematicians to join the society. I am fully aware of the inevitable incompleteness of the overview and I hereby apologize to my numerous colleagues and friends whose work was not directly mentioned in this paper. Finally I want once more to stress the importance of providing courses on fuzzy set theory in the regular curriculum of mathematics, computer science and engineering for the further development of this beautiful and powerful theory.

References [1] Arieli, O., Cornelis, C., Deschrijver, G. and Kerre, E.E. Relating intuitionistic fuzzy sets and interval-valued fuzzy sets through bilattices, in: Applied Computational Intelligence (D. Ruan, P. D’hondt, M. De Cock, M. Nachtegael and E.E. Kerre, eds.), World Scientific, 2004, pp.57-64. [2] Atanassov, K.T. Intuitionistic fuzzy sets, VII ITKR’s Session, Sofia (deposed in Central Sci.-Technical Library of Bulg. Acad. of Sci., 1697/84), 1983, (in Bulgarian). [3] Atanassov, K.T. Intuitionistic fuzzy sets, Physica-Verlag, Heidelberg, 1999. [4] Atanassov, K.T., Deschrijver, G., Cornelis, C., Kerre, E.E. and Orozova, D. Intuitionistic fuzzy modal logic operators: a new outlook, in: Proceedings of the Scientific Conference “Contemporary Technologies”, 2003, pp.32-39. [5] Baczy´ nski, M. and Jayaram, B. Fuzzy implications, Springer, 2008. [6] Bandler, W. and Kohout, L. Fuzzy relational products as a tool for analysis and synthesis of the behavior of complex natural and artificial systems, in: Fuzzy Sets, Theory and Applications (P.P. Wang and S.K. Chang, eds.), Plenum Press, New York, 1980, pp.341-367. [7] Basu, K., Deb, R. and Pattanaik, P.K. Soft sets: an ordinal formulation of vagueness with some applications to the theory of choice, Fuzzy Sets and Systems, V.45, 1992, pp.45-58. [8] Bellman, R. and Giertz, M. On the analytic formalism of the theory of fuzzy sets, Information Sciences, V.5, 1973 pp.149-156. [9] Bilgi¸c, T. and T¨ urk¸sen, B. Measurement of membership functions: theoretical and empirical work, in: Fundamentals of Fuzzy Sets, (D. Dubois and H. Prade, eds.), Kluwer Academic Publishers, 2000, pp.195230. [10] Bodenhofer, U., De Cock, M. and Kerre, E.E. Openings and closures of fuzzy preorderings: theoretical basics and applications to fuzzy rule-based systems, Int. J. General Systems, V.32, 2003, pp.343-360. [11] Bortolan, G. and Degani, R. A review of some methods for ranking fuzzy subsets, Fuzzy Sets and Systems, V.15, 1985, pp.1-19. [12] Bustince, H., Montero, J., Pagola, M., Barrenechea, E. and Gomez, D. A survey of interval-valued fuzzy sets, in: Handbook of Granular Computing, (W. Pedrycz, A. Skowron and V. Kreinovich, eds.), John Wiley and Sons, 2008, pp.491-516. [13] Chang, C.L. Fuzzy topological spaces, J. Math. Anal. Appl., V.24, 1969, pp.182-190. [14] Chen, G., Kerre, E.E. and Vandenbulcke, J. The dependency preserving decomposition and a testing algorithm in a fuzzy relational data model, Fuzzy sets and Systems, V.72, 1995, pp.27-38. [15] Cornelis, C., Atanassov, K.T. and Kerre, E.E. Intuitionistic fuzzy sets and interval-valued fuzzy sets: a critical comparison, in: Proceedings of the 3rd Int. Conf. on Fuzzy Logic and Technology (M. Wagenknecht and Hampel, R. eds.), 2003, pp.159-163.


31

[16] Cornelis, C., De Cock, M. and Kerre, E.E. Intuitionistic fuzzy rough sets at the crossroads of imperfect knowledge, Expert Systems, V.20, 2003, pp.260-270. [17] Cornelis, C., Deschrijver, G., De Cock, M. and Kerre, E.E. Intuitionistic fuzzy relational calculus: an overview, in: Proceedings of First International IEEE Symposium on Intelligent Systems, Varna, V.1, 2002, pp.340-345. [18] Cornelis, C., Deschrijver, G. and Kerre, E.E. Classification of intuitionistic fuzzy implicators: an algebraic approach, in: Proceedings of the 6th Joint Conference on Information Sciences, Durham, 2002, pp.105-108. [19] Cornelis, C., Deschrijver, G. and Kerre, E.E. Implication in intuitionistic fuzzy and interval-valued fuzzy set theory: Construction, classification, application, Int. J. Approx. Reas., V.35, 2004, pp.55-95. [20] Cornelis, C., Deschrijver, G. and Kerre, E.E. Advances and challenges in interval-valued fuzzy logic, Fuzzy Sets and Systems, V.157, 2006, pp. 622–627. [21] Cornelis, C. and Kerre, E.E. Inference in intuitionistic fuzzy expert systems, in:Proceedings of the 6th Int. Conf. on Fuzzy Sets and its Applications, Liptovsk´ y Mikul´ aˇs, 2002, p.45. [22] Cornelis, C. and Kerre, E.E. On the structure and interpretation of an intuitionistic fuzzy expert system, in: Proceedings EUROFUSE, De Baets, B., Fodor, J. and Pasi, G. eds., Varenna, 2002, pp.173-178. [23] Cornelis, C. and Kerre, E.E. Inclusion measures in intuitionistic fuzzy set theory, in: Lecture Notes in Artificial Intelligence, 2003, V.2711, pp.345-356. [24] Cornelis, C., Kerre, E.E. and Atanassov, K.T. Application of extended modal operators as a tool for constructing inclusion indicators over intuitionistic fuzzy sets, in: Soft Computing Foundations and Theoretical Aspects, (Atanassov, K.T., Hryniewicz, O. and Kacprzyk, J. eds.), Akademicka Oficyna Wydawnicza, Warsaw, 2004, pp.133-142. [25] Cornelis, C., Van Der Donck, C. and Kerre, E.E. Sinha-Dougherty approach to the fuzzification of set inclusion revisited, Fuzzy Sets and Systems, V.134, 2003, pp.283-295. [26] De Baets, B. and Kerre, E.E. Fuzzy relational compositions, Fuzzy Sets and Systems, V.60, 1993, pp.109120. [27] De Baets, B. and Kerre, E.E. A revision of B andler-K ohout compositions of relations, Mathematica Pannonica, V.4/1. 1993, pp.1-21. [28] De Baets, B. and Kerre, E.E. Triangular fuzzy relational compositions revisited, in: Uncertainty in Intelligent Systems (Bouchon-Meunier, B., Valverde, L. and Yager, R. eds.), North Holland, Amsterdam, 1993, pp.257268. [29] De Baets, B., Van de Walle, B. and Kerre, E.E. Fuzzy preference structures and their characterization, The Journal of Fuzzy Mathematics, V.3, 1995, pp.373-381. [30] De Baets, B., Van de Walle, B. and Kerre, E.E. Fuzzy preferences structures without incomparability, Fuzzy Sets and Systems, V.76, 1995, pp.333-348. [31] De Baets, B., Van de Walle, B. and Kerre, E.E. A plea for the use of Lukasiewicz triplets in the definition of fuzzy preference structures, Part II: the identity case, Fuzzy sets and Systems, V.99, 1998, pp.303-310. [32] De Cock, M., Cornelis, C. and Kerre, E.E. Fuzzy rough sets: beyond the obvious, in: Proceedings of the IEEE Int. Conf. on Fuzzy Systems, 2004, V.1, pp.103-108. [33] De Cock, M., Cornelis, C. and Kerre, E.E. Fuzzy rough sets: the forgotten step, IEEE Transactions on Fuzzy Systems, V.15, 2007, pp.121-130. [34] De Cock, M. and Kerre, E.E. Fuzzy topologies induced by fuzzy relation based modifiers, in: Computational Intelligence, Theory and applications, (Reusch, B. ed.), Springer, Berlin, 2001, pp.239-248. [35] De Cock, M. and Kerre, E.E. Why fuzzy T -equivalence relations do not resolve the Poincaré paradox and related issues, Fuzzy Sets and Systems, V.133, 2003, pp.175-180. [36] De Cock, M. and Kerre, E.E. Fuzzy modifiers based on fuzzy relations, Information Sciences, V.160, 2004, pp.173-199. [37] De Cock, M., Radzikowska, A.M. and Kerre, E.E. Modelling linguistic modifiers using fuzzy-rough structures, in: Proceedings of the 8th Int. Conf. on Information Processing and Management of Uncertainty in Knowledge-Based Systems, Madrid, 2000, pp.1735-1742. [38] De Cock, M., Radzikowska, A.M. and Kerre, E.E. A fuzzy-rough approach to the representation of linguistic hedges, in: Technologies for Constructing Intelligent Systems, (Bouchon-Meunier, B., Gutiérrez-R´ıos, J., Magdalena, L. and Yager, R. eds.), Springer, 2002, V.89 of Studies in Fuzziness and Soft Computing, pp.33-42. [39] De Cooman, G. The formal analogy between possibility and probability theory, in: Foundations and Applications of Possibility Theory, (de Cooman, G., Ruan, D. and Kerre, E.E. eds.), World Scientific, 1995, pp.71-87. [40] De Cooman, G. Possibility theory I: the measure- and integral-theoretic groundwork, Int. J. General Systems, V.25, 1997, pp.291-323. [41] De Cooman, G. Possibility theory II: conditional possibilities, Int. J. General Systems, V.25, 1997, pp.325351.

32


[42] De Cooman, G. Possibility theory III: possibilistic independence, Int. J. General Systems, V.25, 1997, pp.353-371. [43] Deng, J.L. Introduction to grey system theory, J. Grey Systems, V.1, 1989, pp.1-24. [44] Deschrijver, G., Cornelis, C. and Kerre, E.E. Intuitionistic fuzzy connectives revisited, in: Proceedings of the 9th International Conference on Information Processing and Management of Uncertainty in KnowledgeBased Systems, Annecy, 2002, V.3, pp.1839-1844. [45] Deschrijver, G., Cornelis, C. and Kerre, E.E. On the representation of intuitionistic fuzzy t-norms and t-conorms, IEEE Transactions on Fuzzy Systems, V.12, 2004, pp.45-61. [46] Deschrijver, G. and Kerre, E.E. A generalization of operators on intuitionistic fuzzy sets using triangular norms and conorms, Notes on Intuitionistic Fuzzy Sets, V.8, 2002, pp.19-27. [47] Deschrijver, G. and Kerre, E.E. On the relationship between intuitionistic fuzzy sets and some other extensions of fuzzy set theory, J. Fuzzy Mathematics, V.10, 2002, pp.711-723. [48] Deschrijver, G. and Kerre, E.E. Classes of intuitionistic fuzzy t-norms satisfying the residuation principle, Int. J. Uncertainty, Fuzziness and Knowledge-Based Systems, V.11, 2003, pp. 691-709. [49] Deschrijver, G. and Kerre, E.E. On the cartesian product of intuitionistic fuzzy sets, J. Fuzzy Mathematics, V.11, 2003, pp.537-547. [50] Deschrijver, G. and Kerre, E.E. On the composition of intuitionistic fuzzy relations, Fuzzy Sets and Systems, V.133, 2003, pp.227-235. [51] Deschrijver, G. and Kerre, E.E. On the relationship between some extensions of fuzzy set theory, Fuzzy Sets and Systems, V.133, 2003, pp.227-235. [52] Deschrijver, G. and Kerre, E.E. Uninorms in L∗ -fuzzy set theory, Fuzzy Sets and Systems, V.148, 2004, pp.243-263. [53] Deschrijver, G. and Kerre, E.E. Implicators based on binary aggregation operators in interval-valued fuzzy set theory, Fuzzy Sets and Systems, V.153, 2005, pp.229-248. [54] Deschrijver, G. and Kerre, E.E. On the cuts of intuitionistic fuzzy compositions, Kuwait Journal of Science and Engineering, V.32, 2005, pp.17-38. [55] Deschrijver, G. and Kerre, E.E. Smets-M agrez axioms for R-implications in interval-valued and intuitionistic fuzzy set theory, Int. J. Uncertainty, Fuzziness and Knowledge-Based Systems, V.13, 2005, pp.453-464. [56] Deschrijver, G. and Kerre, E.E. Triangular norms and related operators in L∗ -fuzzy set theory, in: Logical, Analytic and Probabilistic Aspects of Triangular Norms, (Klement,E.P. and Mesiar, R. eds.), Elsevier, 2005, pp.231-259. [57] Deschrijver, G. and Kerre, E.E. On the position of intuitionistic fuzzy set theory in the framework of theories modeling imprecision, Information Sciences, V.177, 2007, pp.1860-1866. [58] Deschrijver, G. and Kerre, E.E. Aggregation operators in interval-valued fuzzy and A tanassov’s intuitionistic fuzzy set theory, in: Fuzzy Sets and their Extensions: Representation, Aggregation and Models (Bustince, H., Herrera, F. and Montero, J. eds.), Springer-Verlag, Studies in Fuzziness and Soft Computing, V.220, 2008, pp.183-203. [59] Dubois, D., Kerre, E.E., Mesiar, R. and Prade, H. Fuzzy interval analysis, in: Fundamentals of fuzzy sets, (Dubois, D. and Prade, H. eds.), Kluwer Academic Publishers, 2000, pp.483-582. [60] Dubois, D. and Prade, H. Twofold fuzzy sets and rough sets–Some issues in knowledge representation, Fuzzy Sets and Systems, V.23, 1987, pp.3-18. [61] Dubois, D. and Prade, H. Rough fuzzy sets and fuzzy rough sets, Int. J.General Systems, V.17, 1990, pp.191-209. [62] Dubois, D. and Prade, H. Toll sets and toll logic, in: fuzzy logic: State of the Art, (Lowen, R. and Roubens, M. eds.), Kluwer Academic Publishers, 1993, pp.169-177. [63] Gaines, B. Foundations of fuzzy reasoning, in: Fuzzy automata and decision processes, (Gupta, M., Saridis, G. and Gaines,B. eds.), North Holland, 1977, pp.19-75. [64] Gau, W.L. and Buehrer, D.J. Vague sets, IEEE Transactions Systems, Man, Cybernetics, V.23, 1993, pp.610-614. [65] Gentilhomme, Y. Les sous-ensembles flous en linguistique, Cahiers de Linguistique Théorique et Appliquée, V, 1968, pp.47-63. [66] Goguen, J. L-fuzzy sets, J. Math. Anal. Appl., V.18, 1967, pp.145-174. [67] Groenemans, R., Kerre, E.E. and Van Ranst, E. Fuzzy relational calculus in land evaluation, Geoderma, V.77, 1997, pp.283-298. [68] Hirota, K. Concepts of probabilistic sets, Fuzzy Sets and Systems, V.5, 1981, pp.31-46. [69] Kandil, A., Kerre E.E. and Nouh, A. Operations and mappings on fuzzy topological spaces, Annales de la Société Scientifique de Bruxelles, V.105, 1991, pp.167-188. [70] Katsaras, A.K. and Liu, D.B. Fuzzy vector spaces and fuzzy topological vector spaces, J. Math. Anal. Appl., V.58, 1977, pp.135-146.


33

[71] Kerre, E.E. A tribute to Zadeh’s extension principle, Invited paper for the special issue of Iranica Scientia Journal for Zadeh’s 90th birthday, to appear in 2011. [72] Kerre, E.E. Characterization of normality in fuzzy topological spaces, Simon Stevin, V.53, 1979, pp.239-248. [73] Kerre, E.E. Fuzzy Sierpinski space and its generalizations, J. Math. Anal. Appl., V.74, 1980, pp.318-324. [74] Kerre, E.E. A walk through fuzzy relations and their application to information retrieval, medical diagnosis and expert systems, in: Proceedings of ISUMA ’90 (Ayyub, B. ed.), IEEE Computer Society Press, Los Alamitos, 1990, pp.84-89. [75] Kerre, E.E. Introduction to the basic principles of fuzzy set theory and some of its applications, Communication and Cognition, Gent, 1993, ISBN 90-70963-41-8. [76] Kerre, E.E. Fuzzy sets and approximate reasoning (english edition), Xian Jiaotong University Press, 1999, ISBN 7-5605-1091-4/O137. [77] Kerre, E.E. A first view on the alternatives of fuzzy set theory, in: Computational Intelligence in Theory and Practice, (Reusch, B. and Temme, K.-H. eds.), Physica-Verlag, Heidelberg, 2001, pp.55-72. [78] Kerre, E.E., Nouh, A. and Kandil, A. Operations on the class of fuzzy sets on a universe endowed with a fuzzy topology, J. Math. Anal. Appl., V.180, 1993, pp.325-341. [79] Kerre, E.E. and Ottoy, P. On the characterization of a Chang fuzzy topology by means of a Kerre neighborhood system, in: Proceedings NAFIPS 87, (Chameau, J.L. and Yao, J. eds.), Purdue University Press, 1987, pp.302-307. [80] Kerre, E.E. and Ottoy, P. On the different notions of neighborhood in Chang-Goguen fuzzy topological spaces, Simon Stevin, V.61, 1987, pp.131-146. [81] Kerre, E.E. and Ottoy, P. A comparison of the different notions of neighborhood systems for Chang topologies, in: Proceedings of the First Joint IFSA-EC EURO – WG Workshop on Progress in Fuzzy Sets in Europe, (Kacprzyk, J. and Straszak, A. eds.), Prace IBS PAN, Warsaw, 1989, V.169, pp.241-251. [82] Kerre, E.E. and Ottoy, P. Lattice properties of neighborhood systems in Chang fuzzy topological spaces, Fuzzy Sets and Systems, V.30, 1989, pp.205-213. [83] Kerre, E.E. Zenner, R. and De Caluwe, R. The use of fuzzy set theory in information retrieval and databases: a survey, J. American Society for Information Science, V.37, 1986, pp.341-345. [84] Kitainik, L. Fuzzy decision procedures with binary relations, kluwer academic publishers, Dordrecht, 1993. [85] Kuijken, L., Van Maldeghem, H. and Kerre, E.E. Fuzzy projective geometries from fuzzy groups, Tatra Mt. Math. Publ., V.16, 1999, pp.95-108. [86] Lowen, R. Fuzzy topological spaces and fuzzy compactness, J. Math. Anal. Appl., V.56, 1976, pp.621-633. [87] Martens, J., Put, F. and Kerre, E.E. A fuzzy-neural resemblance approach to validate simulation models, Soft Computing, V.11, 2007, pp.299-307. [88] Mélange, T., Nachtegael, M., Sussner, P. and Kerre, E.E. On the decomposition of interval-valued fuzzy morphological operations, J. Mathematical Imaging and Vision, V.36, 2010, pp.270-290. [89] Mizumoto, M. and Tanaka, K. Some properties of fuzzy numbers, in: Advances in fuzzy set theory and applications, (Gupta, M. Ragade, R. and Yager, R. eds.), North Holland, 1979, pp.153-164. [90] Mordeson, J., Bhutani, K. and Rosenfeld, A. Fuzzy group theory, Springer, 2005. [91] Mordeson, J., Malik, D. and Kuroki, N. Fuzzy Semigroups, Springer, 2003. [92] Nachtegael, M., De Cock, M., Vander Weken, D. and Kerre, E.E. Fuzzy relational images in computer science, in: Relational Methods in Computer Science, (de Swart, H.M. ed.), Springer, Berlin, Lecture Notes in Computer Science, 2002, pp.134-151. [93] Nachtegael, M., Radzikowska, A.M. and Kerre, E.E. On links between fuzzy morphology and fuzzy rough sets, in:Proceedings of the 8th Int. Conf. on Information Processing and Management of Uncertainty in Knowledge-Based Systems, Madrid, 2000, pp.1381-1388. [94] Negoita, C. and Ralescu, D. Application of fuzzy sets to system analysis, Birkhauser Verlag, Basel, 1975. [95] Pawlak, Z. Rough sets and fuzzy sets, Fuzzy Sets and Systems, V.17, 1985, pp.99-102. [96] Radzikowska, A.M. and Kerre, E.E. Fuzzy rough sets revisited, in:Proceedings of the 7th EUFIT Conference (CDROM), Aachen, 1999. [97] Radzikowska, A.M. and Kerre, E.E. A comparative study of fuzzy rough sets, Fuzzy Sets and Systems, V.126, 2002, pp.137-156. [98] Radzikowska, A.M. and Kerre, E.E. Fuzzy rough sets based on residuated lattices, in: Transactions on Rough Sets II: Rough Sets and Fuzzy Sets, V. 3135 of Lecture Notes in Computer Science, 2004, pp.278-297. [99] Radzikowska, A.M. and Kerre, E.E. On L-fuzzy rough sets, in: Artificial Intelligence and Soft Computing, V.3070 of Lecture Notes in Artificial Intelligence, 2004, pp.526-531. [100] Rosenfeld, A. Fuzzy groups, J. Math. Anal. Appl., V.35, 1971, pp.512-517. [101] Sambuc, R. Fonctions Φ-floues. Application ` a l’aide au diagnostic en pathologie thyroidienne, Ph.D. thesis, Univ. Marseille, 1975. [102] Schmidt, G. Relational mathematics, V.132 of Series Encyclopedia of Mathematics and its Applications, Cambridge University Press, 2011.

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[103] Schockaert, S., De Cock, M. and Kerre, E.E. Fuzzifying Allen’s temporal interval relations, IEEE Transactions on Fuzzy Systems, V.16, 2008, pp.517-533. [104] Schockaert, S., De Cock, M. and Kerre, E.E. Spatial reasoning in a fuzzy region connection calculus, Artificial Intelligence, V.173, 2009, pp.258-298. [105] Schweizer, B. and Sklar, A. Associative functions and statistical triangle inequalities, Publ. Math. Debrecen, V.8, 1961, pp.169-186. [106] Shi, Y., Van Gasse, B., Ruan, D. and Kerre, E.E. On the first place antitonicity in QL-implications, Fuzzy Sets and Systems, V.159, 2008, pp.2988-3013. [107] Shi, Y., Van Gasse, B., Ruan, D. and Kerre, E.E. On dependencies and independencies of fuzzy implication axioms, Fuzzy Sets and Systems, V.161, 2010, pp.1388-1405. [108] Sinha, D. and Dougherty, E.R. Fuzzification of set inclusion: theory and applications, Fuzzy Sets and Systems, V.55, 1993, pp.15-42. [109] Smets, P. and Magrez, P. Implication in fuzzy logic, Int. J. Approx. Reas., V.1, 1987, pp.327-347. [110] Sugeno, M. Theory of fuzzy integrals and its applications, Ph.D. thesis, Tokyo Institute of Technology, 1974. [111] Van de Walle, B., De Baets, B. and Kerre, E.E. A plea for the use of Lukasiewicz triplets in the definition of fuzzy preference structures. Part I: general argumentation, Fuzzy Sets and Systems, N.97, 1998, pp.349-359. [112] Van Leekwijck, W. and Kerre, E.E. Defuzzification: criteria and classification, Fuzzy Sets and Systems, V.108, 1999, pp.159-178. [113] Walker, C. and Walker, E. The algebra of truth values of type-2 fuzzy sets: a survey, in: Interval/Probabilistic Uncertainty and Non-classical Logos, (Huynh, V.N. Nakamori, Y., Ono, H., Lawry, J., Kreinovich, V. and Nguyen, H.T. eds.), Springer, V.46 of Advances in Soft Computing, 2008, pp.245-255. [114] Walley, P. Statistical reasoning with imprecise probabilities, chapman & hall, 1991. [115] Wang, X. and Kerre, E.E. Some relationships between different ordering procedures based on a fuzzy relation, in: Proceedings 7th IPMU conference, Paris, 1998, pp.250-254. [116] Wang, X. and Kerre, E.E. Reasonable properties for the ordering of fuzzy quantities I and II, Fuzzy Sets and Systems, V.118, 2001, pp.375-385, pp.387-405. [117] Zadeh, L. Fuzzy sets, Information and Control, V.8, 1965, pp.338-353. [118] Zadeh, L.A. Similarity relations and fuzzy orderings, Information Sciences, V.3, 1971, pp.177-200. [119] Zadeh, L.A. The concept of a linguistic variable and its application to approximate reasoning I, Information Sciences, V.8, 1975, pp.199-249. [120] Zadeh, L.A. Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems, V.1, 1978, pp.3-28. [121] Zhang, W.-R. Bipolar fuzzy sets and relations: a computational framework for cognitive modeling and multi-agent decision analysis, in: Proceedings of the 1st Int. Joint Conf. of the North American Fuzzy Information Processing Society (NAFIPS/IFIS/NASA0), 1994, pp.305-309. [122] Zhang, W.-R. YinYang bipolar relativity: a Unifying Theory of Nature, Agents and Causality with Applications in Quantum Computing, Cognitive Informatics and Life Sciences, IGI Global, 2011. [123] Zimmermann, H.J. and Zysno, P. Latent connectives in human decision making, Fuzzy Sets and Systems, V.4, 1980, pp.37-51.

Etienne E. Kerre -was born in Zele, Belgium on May 8, 1945. He obtained his M.Sc. degree in Mathematics in 1967 and his Ph.D. in Mathematics in 1970 from Ghent University. Since 1984, he has been a lector, and since 1991, a full professor at Ghent University. He is a referee for 65 international scientific journals, and also a member of the editorial board of international journals and conferences on fuzzy set theory. He was an honorary chairman at various international conferences. In 1976, he founded the Fuzziness and Uncertainty Modeling Research Unit (FUM) and since then his research has been focused on the modeling of fuzziness and uncertainty, and has resulted in a great number of contributions in fuzzy set theory and its various generalizations. Especially the theories of fuzzy relational calculus and of fuzzy mathematical structures owe a very great deal of him. Over the years he has also been a promotor of 29 Ph.D’s on fuzzy set theory. His current research interests include fuzzy and intuitionistic fuzzy relations, fuzzy topology, and fuzzy image processing. He has authored or co-authored 25 books, and more than 450 papers appeared in international refereed journals and proceedings.He has been awarded the IFSA fellowship in 2007.