Montero, Javier and Gómez, D. and Bustince, H.
(2007)
*On the relevance of some families of fuzzy Sets.*
Fuzzy Sets and Systems, 158
(22).
pp. 2429-2442.
ISSN 0165-0114

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Official URL: http://www.sciencedirect.com/science/article/pii/S0165011407002126

## Abstract

In this paper we stress the relevance of a particular family of fuzzy sets, where each element can be viewed as the result of a classification problem. In particular, we assume that fuzzy sets are defined from a well-defined universe of objects into a valuation space where a particular graph is being defined, in such a way that each element of the considered universe has a degree of membership with respect to each state in the valuation space. The associated graph defines the structure of such a valuation space, where an ignorance state represents the beginning of a necessary learning procedure. Hence, every single state needs a positive definition, and possible queries are limited by such an associated graph. We then allocate this family of fuzzy sets with respect to other relevant families of fuzzy sets, and in particular with respect to Atanassov's intuitionistic fuzzy sets. We postulate that introducing this graph allows a natural explanation of the different visions underlying Atanassov's model and interval valued fuzzy sets, despite both models have been proven equivalent when such a structure in the valuation space is not assumed.

Item Type: | Article |
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Uncontrolled Keywords: | Atanassov’s intuitionistic fuzzy sets; Interval valued fuzzy sets; Type-2 fuzzy sets; L-fuzzy sets |

Subjects: | Sciences > Mathematics > Logic, Symbolic and mathematical |

ID Code: | 16327 |

References: | A. Amo, D. Gómez, J. Montero, G. Biging, Relevance and redundancy in fuzzy classification systems, Mathware Soft Computing 8 (2001)203–216. A. Amo, J. Montero, G. Biging, V. Cutello, Fuzzy classification systems, Eur. J. Oper. Res. 156 (2004) 459–507. A. Amo, J. Montero, E. Molina, Representation of consistent recursive rules, Eur. J. Oper. Res. 130 (2001) 29–53. K.T. Atanassov, Intuitionistic fuzzy sets, in: V. Sgurev (Ed.), VII ITKR’s Session, Sofia, June 1983 (deposed in Central Science and Technical Library, Bulgarian Academy of Sciences, 1697/84, in Bulgarian). K.T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986) 87–96. K.T. Atanassov, More on intuitionistic fuzzy sets, Fuzzy Sets and Systems 33 (1989) 37–45. K.T. Atanassov, Intuitionistic Fuzzy Sets, Physica-Verlag, Heidelberg, NewYork, 1999. [8] K.T. Atanassov, Answer to D. Dubois, S. Gottwald, P. Hajek, J. Kacprzyk and H. Prade’s paper, Terminological difficulties in fuzzy set theory—the case of intuitionistic fuzzy sets, Fuzzy Sets and Systems 156 (2005) 496–499. K.T. Atanassov, My personal view on intuitionistic Fuzzy sets theory. In: H. Bustince et al. (Eds.), Fuzzy Sets and Their Extensions:Representation, Aggregation and Models, Studies in Fuzziness and Soft Computing (Studfuzz) 220, Springer, Berlin, New York, 2007,pp. 25–46. [10] J.C. Bezdek, J.D. Harris, Fuzzy partitions and relations: an axiomatic basis for clustering, Fuzzy Sets and Systems 1 (1978) 111–127. H. Bustince, P. Burillo, Structures on intuitionistic fuzzy relations, Fuzzy Sets and Systems 78 (1996) 293–303. H. Bustince, P. Burillo, Vague sets are intuitionistic fuzzy sets, Fuzzy Sets and Systems 79 (1996) 403–405. H. Bustince, J. Montero, M. Pagola, E. Barrenechea, D. Gómez, A survey on interval-valued fuzzy sets, in: W.Pedrycz, A. Skowron, V. Kreinovichedrycz (Eds.), Handbook of Granular Computing,Wiley, NewYork, 2007, in press. D. Butnariu, Additive fuzzy measures and integrals, J.Math. Anal. Appl. 93 (1983) 436–452. T. Calvo, A. Kolesarova, M. Komornikova, R. Mesiar,Aggregation operators: properties, classes and construction methods, in: T. Calvo, G. Mayor, R. Mesiar (Eds.), Aggregation Operators, Springer, NewYork, 2002, pp. 3–104. G. Cattaneo, D. Ciucci, Basic intuitionistic principles in fuzzy set theories and its extensions (a terminological debate on Atanassov IFS), Fuzzy Sets and Systems 157 (2006) 3198–3219. P. Cintula, Basics of a formal theory of fuzzy partitions, in: Proc. EUSFLAT’05 Conf., Technical University of Catalonia, Barcelona, 2005,pp. 884–888. C. Cornelis, G. Deschrijver, E.E. Kerre, Implication in intuitionistic fuzzy and interval-valued fuzzy set theory: construction, classification,application, Int. J.Approximate Reasoning 35 (2004) 55–95. C. Cornelis, G. Deschrijver, E.E. Kerre, Advances and challenges in interval-valued fuzzy logic, Fuzzy Sets and Systems 157 (2006) 622–627. S. Coupland, R. John, Type-2 fuzzy logic and the modelling of uncertainty. In: H. Bustince et al. (Eds.), Fuzzy Sets and Their Extensions:Representation, Aggregation and Models, Studies in Fuzziness and Soft Computing (Studfuzz) 220, Springer, Berlin, New York, 2007,pp. 3–23. V. Cutello, J. Montero, Recursive connective rules, Int. J. Intell. Syst. 14 (1999) 3–20. B. De Baets, R. Mesiar, T-partitions, Fuzzy Sets and Systems 97 (1998) 211–223. G. Deschrijver, C. Cornelis, E.E. Kerre, On the representation of intuitionistic fuzzy t-norms and t-conorms, IEEE Trans. Fuzzy Systems 12 (2004) 45–61. G. Deschrijver, E.E. Kerre, On the relationship between some extensions of fuzzy sets theory, Fuzzy Sets and Systems 133 (2003) 227–235. D. Dubois, S. Gottwald, P. Hajek, J. Kacprzyk, H. Prade, Terminological difficulties in fuzzy set theory—the case of intuitionistic fuzzy sets,Fuzzy Sets and Systems 156 (2005) 485–491. D. Dumitrescu, Fuzzy partitions with the connectives T∞, S∞, Fuzzy Sets and Systems 47 (1992) 193–195. P. Fortemps, R. Slowinski,Agraded quadrivalent logic for ordinal preference modelling: loyola-like approach, Fuzzy Optimization and Decision Making 1 (2002) 93–111. J. Goguen, L-fuzzy sets, J. Math. Ann. Appl. 18 (1967) 145–174. D. Gómez, J. Montero, A discussion on aggregation operators, Kybernetika 40 (2004) 107–120. D. Gómez, J. Montero, G. Biging, Accuracy measures for fuzzy classification in remote sensing, in: Proc. IPMU’06 Conf., Editions E.D.K.,Paris, 2006, pp. 1556–1563. D. Gómez, J. Montero, J.Yá nez, A coloring algorithm for image classification, Inf. Sci. 176 (2006) 3645–3657. D. Gómez, J. Montero, J. Yá nez, C. Poidomani, A graph coloring approach for image segmentation, OMEGA—Internat. J. Manage. Sci. 35 (2007) 173–183. J. González-Pachón, D. Gómez, J. Montero, J.Yánez, Soft dimension theory, Fuzzy Sets and Systems 137 (2003) 137–149. J. González-Pachón, D. Gómez, J. Montero, J.Yánez,Searching for the dimension of binary valued preference relations, Internat. J.Approximate Reasoning 33 (2003) 133–157. I. Iancu, Connectives for fuzzy partitions, Fuzzy Sets and Systems 101 (1999) 509–512. A. Kaufmann, M.M. Gupta, Introduction to Fuzzy Arithmetic, Van Nostrand Reinhold Co., NewYork, 1985. E.E. Kerre, A deeper look on fuzzy numbers from a theoretical as well as from a practical point of view, in: M.M. Gupta, T.Yamakawa (Eds.),Fuzzy Logic in Knowledge-Based Systems, Decision and Control, Elsevier, Amsterdam, 1979, pp. 153–164. C.S. Kim, D.S. Kim, J.S. Park, A new fuzzy resolution principle based on the antonym, Fuzzy Sets and Systems 113 (2000) 299–307. E.P. Klement, R. Mesiar, E. Pap, Triangular Norms, Kluwer Academic Publishers, Dordrecht, 2002. S. Kundu, J. Chen, Fuzzy logic or Lukasiewicz logic: a clarification, Fuzzy Sets and Systems 95 (1998) 369–379. H. Lee-Kwang, J.H. Lee, A method for ranking fuzzy numbers and its application to decision-making, IEEE Trans. Fuzzy Systems 7 (1999)677–685. J.M. Mendel, Advances in type-2 fuzzy sets and systems, Inform. Sci. 117 (2007) 84–110. J.M. Mendel, R.I. John, Type-2 fuzzy sets made simple, IEEE Trans. Fuzzy Systems 10 (2002) 117–127. H.B. Mitchell, Ranking type-2 fuzzy numbers, IEEE Trans. Fuzzy Systems 14 (2006) 287–294. J. Montero, Comprehensive fuzziness, Fuzzy Sets and Systems 20 (1986) 86–89. J. Montero, Extensive fuzziness, Fuzzy Sets and Systems 21 (1987) 201–209. J. Montero, D. Gómez, Preferences, classification and intuitionistic fuzzy sets, in: Proc. EUSFLAT’03 Conf. University of Applied Sciences,Zittau, 2003, pp. 187–192. J. Montero, M. Mendel, Crisp acts, fuzzy decisions, in: S. Barro et al. (Eds.),Advances in Fuzzy Logic, Universidad de Santiago de Compostela,Santiago de Compostela, 1998, pp. 219–238. A.M. Norwich, L.B. Turksen, A model for the measurement of membership and the consequences of its empirical implementation, Fuzzy Sets and Systems 12 (1984) 1–25. V. Novak, Antonyms and linguistic quantifiers in fuzzy logic, Fuzzy Sets and Systems 124 (2001) 335–351. C. Paradis, C.Willners, Antonymy and negation—the boundness hypothesis, J. Pragmatics 38 (2006) 1051–1080. J.A. Robinson, A machine oriented logic based on the resolution principle, J. ACM 12 (1965) 23–41. E.H. Ruspini, A new approach to clustering, Inform. Control 15 (1969) 22–32. R. Sambuc, Function -flous, application a l’aide au diagnostic en pathologie thyroidienne, These de doctorat en Medicine, Marseille, 1975. G. Shafer, Savage revisited, Statist. Sci. 1 (1986) 463–501. E. Szmidt, J. Kacprzyk, Distances between intuitionistic fuzzy sets, Fuzzy Sets and Systems 114 (2000) 505–518. G. Takeuti, S. Titani, Intuitionistic fuzzy logic and intuitionistic fuzzy set theory, J. Symblic Logic 49 (1984) 851–866. E. Trillas, Sobre funciones de negación en la teoría de los subconjuntos difusos, Stochastica III-1 (1979) 47–59 (in Spanish). English version in S. Barro et al. (Eds.) (1998): Advances of Fuzzy Logic, Universidad de Santiago de Compostela, Santiago de Compostela, pp. 31–43. E. Trillas, On the use of words and fuzzy sets, Inform. Sci. 176 (2006) 1463–1487. I.B. Türksen, Interval valued fuzzy sets based on normal forms, Fuzzy Sets and Systems 20 (1986) 191–210. R.R. Yager, On ordered averaging aggregation operators in multicriteria decision making, IEEE Trans. Systems Man Cybernet. 18 (1988)183–190. R.R.Yager, A. Rybalov, Uninorm aggregation operators, Fuzzy Sets and Systems 80 (1996) 111–120. L.A. Zadeh, Fuzzy sets, Inform. Control 8 (1965) 338–353. L.A. Zadeh, The concept of linguistic variable and its application to approximate reasoning I, Inform. Sci. 8 (1975) 199–249. L.A. Zadeh, The concept of linguistic variable and its application to approximate reasoning II, Inform. Sci. 8 (1975) 301–357. L.A. Zadeh, The concept of linguistic variable and its application to approximate reasoning III, Inform. Sci. 9 (1975) 43–80. L.A. Zadeh, J. Kacprzyk (Eds.), Computing withWords in Information-intelligent Systems, (2 vols.) Physica Verlag, Heidelberg, 1999. |

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