×

Intuitionistic fuzzy generators. Application to intuitionistic fuzzy complementation. (English) Zbl 0961.03051

The concept of an intuitionistic fuzzy generator \(\varphi: [0,1]\to[0,1]\) is introduced by \(\varphi(x)\leq 1-x\) for all \(x\in[0,1]\). It can be continuous, decreasing or increasing. Some properties of the new concept are studied and characterization theorems for it are proved. One of the most important results is:
Theorem 25. Let \(A=\{\langle x,\mu_A(x) \rangle:x\in X\}\) be a fuzzy set on the referential set \(X\neq\emptyset\), and let \(\varphi\) be an intuitionistic fuzzy generator. The set \(\overline A=\{\langle x,\mu_A (x), \varphi(\mu_A(x)) \rangle:x\in X\}\) is an intuitionistic fuzzy set on \(X\) [see K. T. Atanassov, “Intuitionistic fuzzy sets”, Fuzzy Sets Syst. 20, 87-96 (1986; Zbl 0631.03040); Intuitionistic fuzzy sets. Theory and applications. Physica-Verlag, Heidelberg (1999; Zbl 0939.03057)].

MSC:

03E72 Theory of fuzzy sets, etc.
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Alsina, C.; Trillas, E.; Valverde, L., On some logical connectives for fuzzy sets theory, J. Math. Ann. Appl., 93, 15-26 (1983) · Zbl 0522.03012
[2] Atanassov, K., Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20, 87-96 (1986) · Zbl 0631.03040
[3] Atanassov, K., Review and new results on intuitionistic fuzzy sets, IM-MFAIS, 1 (1988)
[4] Atanassov, K., Two operators on intuitionistic fuzzy sets, Comptes rendus de l’Académi bulgare des Sciences, Tome, 41, 5 (1988)
[5] Bellman, R.; Giertz, M., On the analytic formalism of the theory of fuzzy sets, Inform. Sci., 5, 149-156 (1973) · Zbl 0251.02059
[6] Burillo, P.; Bustince, H., Construction theorems for intuitionistic fuzzy sets, Fuzzy Sets and Systems, 84, 271-281 (1996) · Zbl 0903.04001
[7] Burillo, P.; Bustince, H., Entropy on intuitionistic fuzzy sets and on interval-valued fuzzy sets, Fuzzy Sets and Systems, 78, 305-316 (1996) · Zbl 0872.94061
[8] H. Bustince, Conjuntos Intuicionistas e Intervalo valorados Difusos: Propiedades y Construcción. Relaciones Intuicionistas Fuzzy. Thesis, Universidad Pública de Navarra, 1994.; H. Bustince, Conjuntos Intuicionistas e Intervalo valorados Difusos: Propiedades y Construcción. Relaciones Intuicionistas Fuzzy. Thesis, Universidad Pública de Navarra, 1994.
[9] Bustince, H.; Burillo, P., Correlation of interval-valued intuitionistic fuzzy sets, Fuzzy Sets and Systems, 74, 237-244 (1995) · Zbl 0875.94156
[10] Bustince, H.; Burillo, P., Vague sets are intuitionistic fuzzy sets, Fuzzy Sets and Systems, 79, 403-405 (1996) · Zbl 0871.04006
[11] F. Esteva, E. Trillas, X. Domingo, Weak and strong negation function for fuzzy set theory, Proc. 11th IEEE Internat. Symp. on Multivalued Logic, Norman, Oklahoma, 1981, pp. 23-27.; F. Esteva, E. Trillas, X. Domingo, Weak and strong negation function for fuzzy set theory, Proc. 11th IEEE Internat. Symp. on Multivalued Logic, Norman, Oklahoma, 1981, pp. 23-27. · Zbl 0548.03036
[12] Gau, W. L.; Buehrer, D. J., Vague sets, IEEE Trans. Systems Man Cybernet., 23, 2, 610-614 (1993) · Zbl 0782.04008
[13] Higashi, M.; Klir, G. J., On measure of fuzziness and fuzzy complements, Internat. J. General Systems, 8, 3, 169-180 (1982) · Zbl 0484.94047
[14] Lowen, R., On fuzzy complements, Inform. Sci., 14, 2, 107-113 (1978) · Zbl 0416.03047
[15] Ovchinnikov, S. V., Structure of fuzzy binary relations, Fuzzy Sets and Systems, 6, 2, 169-195 (1981) · Zbl 0464.04004
[16] Ovchinnikov, S. V., General negations in fuzzy set theory, J. Math. Anal. Appl., 92, 1, 234-239 (1983) · Zbl 0518.04003
[17] Chem, Shyi-Ming; Tan, Jiann-Mean, Handling multicriteria fuzzy decision-making problems based on vague set theory, Fuzzy Sets and Systems, 67, 163-172 (1994) · Zbl 0845.90078
[18] M. Sugeno, Fuzzy measures and fuzzy integrals: a survey, in: M.M. Gupta, G.N. Saridis, B.R. Gaines (Eds.), Fuzzy Automata and Decision Processes, North-Holland, Amsterdam, 1977, pp. 89-102. Institute of Technology (1974).; M. Sugeno, Fuzzy measures and fuzzy integrals: a survey, in: M.M. Gupta, G.N. Saridis, B.R. Gaines (Eds.), Fuzzy Automata and Decision Processes, North-Holland, Amsterdam, 1977, pp. 89-102. Institute of Technology (1974).
[19] Sugeno, M.; Terano, T., A model of learning based on fuzzy information, Kybernetes, 6, 3, 157-166 (1977)
[20] Szmidt, E.; Kacprzyk, J., Intuitionistic fuzzy sets in group decision making, Notes IFS, 2, 15-32 (1996)
[21] E. Trillas, Sobre funciones de negacion en la teoria de conjuntos difusos, Stochastica III-1 (1979) 47-59.; E. Trillas, Sobre funciones de negacion en la teoria de conjuntos difusos, Stochastica III-1 (1979) 47-59.
[22] E. Trillas, C. Alsina, J.M. Terricabras, Introducción a la lógica borrosa, Ariel Matemática (1995).; E. Trillas, C. Alsina, J.M. Terricabras, Introducción a la lógica borrosa, Ariel Matemática (1995).
[23] Weber, S., A general concept of fuzzy connectives, negations and implications based on t-norms and t-conorms, Fuzzy Sets and Systems, 11, 115-134 (1983) · Zbl 0543.03013
[24] Yager, R. R., On the measure of fuzziness and negation. Part I: membership in the unit interval, Internat. J. General Systems, 5, 221-229 (1979) · Zbl 0429.04007
[25] Yager, R. R., On the measure of fuzziness and negation. Part II: Lattices, Inform. and Control, 44, 3, 236-260 (1979) · Zbl 0429.04008
[26] Zadeh, L. A., Similarity relations and fuzzy orderings, Inform. Sci., 3, 177-200 (1971) · Zbl 0218.02058
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.