Article
Full entry |
PDF
(1.7 MB)
Feedback
PDF
(1.7 MB)
Feedback
Keywords:
t-norm; T-conorm; finite chain; smoothness; implication operator
t-norm; T-conorm; finite chain; smoothness; implication operator
Summary:
This paper is devoted to the study of two kinds of implications on a finite chain $L$: $S$-implications and $R$-implications. A characterization of each kind of these operators is given and a lot of different implications on $L$ are obtained, not only from smooth t-norms but also from non smooth ones. Some additional properties on these implications are studied specially in the smooth case. Finally, a class of non smooth t-norms including the nilpotent minimum is characterized. Any t-norm in this class satisfies that both, its $S$-implication and its $R$-implication, agree.
References:
[1] Baets B. De: Model implicators and their characterization. In: Proc. First ICSC Internat. Symposium on Fuzzy Logic, Zürich, Switzerland (N. Steele, ed.). ICSC Academic Press 1995, pp. A42–A49
[2] Baets B. De, Fodor J. C.: On the structure of uninorms and their residual implicators. In: Proc. 18th Linz Seminar on Fuzzy Set Theory, Linz, Austria 1997, pp. 81–87
[3] Baets B. De, Fodor J. C.: Residual operators of representable uninorms. In: Proc. Fifth European Congress on Intelligent Techniques and Soft Computing, Volume 1 (H. J. Zimmermann, ed.), ELITE, Aachen, Germany 1997, pp. 52–56
[4] Baets B. De, Mesiar R.: Residual implicators of continuous t-norms. In: Proc. Fourth European Congress on Intelligent Techniques and Soft Computing, Volume 1 (H. J. Zimmermann, ed.), ELITE, Aachen, Germany 1997, pp. 27–31
[5] Baets B. De, Mesiar R.: Triangular norms on product lattices. Fuzzy Sets and Systems 104 (1999), 61–75 DOI 10.1016/S0165-0114(98)00259-0 | MR 1685810 | Zbl 0935.03060
[6] Bustince H., Burillo, P., Soria F.: Automorphisms, negations and implication operators. Fuzzy Sets and Systems 134 (2003), 209–229 DOI 10.1016/S0165-0114(02)00214-2 | MR 1969102
[7] Cignoli R., Esteva F., Godo, L., Montagna F.: On a class of left-continuous t-norms. Fuzzy Sets and Systems 131 (2002), 283–296 DOI 10.1016/S0165-0114(01)00215-9 | MR 1939841
[8] Cignoli R., Esteva F., Godo, L., Torrens A.: Basic fuzzy logic is the logic of continuous t-norms and their residua. Soft Computing 4 (2000), 106–112 DOI 10.1007/s005000000044
[9] Cignoli R., D’Ottaviano, I., Mundici D.: Algebraic Foundations of Many-valued Reasoning. Kluwer, Dordrecht 2000 MR 1786097 | Zbl 0937.06009
[10] Esteva F., Godo L.: Monoidal t-norm based logic: towards a logic for left-continuous t-norms. Fuzzy Sets and Systems 124 (2001), 271–288 MR 1860848 | Zbl 0994.03017
[11] Fodor J. C.: On fuzzy implication operators. Fuzzy Sets and Systems 42 (1991), 293–300 DOI 10.1016/0165-0114(91)90108-3 | MR 1127976 | Zbl 0736.03006
[12] Fodor J. C.: Contrapositive symmetry of fuzzy implications. Fuzzy Sets and Systems 69 (1995), 141–156 DOI 10.1016/0165-0114(94)00210-X | MR 1317882 | Zbl 0845.03007
[13] Fodor J. C.: Smooth associative operations on finite ordinal scales. IEEE Trans. Fuzzy Systems 8 (2000), 791–795 DOI 10.1109/91.890343
[14] Godo L., Sierra C.: A new approach to connective generation in the framework of expert systems using fuzzy logic. In: Proc. XVIIIth ISMVL, Palma de Mallorca, Spain 1988, pp. 157–162
[15] Hájek P.: Mathematics of Fuzzy Logic. Kluwer, Dordrecht 1998
[16] Hájek P.: Basic fuzzy logic and BL-algebras. Soft Computing 2 (1998), 124–128 DOI 10.1007/s005000050043
[17] Jenei S.: New family of triangular norms via contrapositive symmetrization of residuated implications. Fuzzy Sets and Systems 110 (2000), 157–174 MR 1747749 | Zbl 0941.03059
[18] Mas M., Mayor, G., Torrens J.: $t$-operators and uninorms on a finite totally ordered set. Internat. J. Intelligent Systems (Special Issue: The Mathematics of Fuzzy Sets) 14 (1999), No. 9, 909–922 DOI 10.1002/(SICI)1098-111X(199909)14:9<909::AID-INT4>3.0.CO;2-B | MR 1691482 | Zbl 0948.68173
[19] Mas M., Monserrat, M., Torrens J.: On left and right uninorms on a finite chain. Fuzzy Sets and Systems, forthcoming MR 2074199 | Zbl 1045.03029
[20] Mas M., Monserrat, M., Torrens J.: Operadores de implicación en una cadena finita. In: Proc. Estylf-2002, León, Spain 2002, pp. 297–302
[21] Mas M., Monserrat, M., Torrens J.: On some types of implications on a finite chain. In: Proc. Summer School on Aggregation Operators 2003 (AGOP’2003), Alcalá de Henares, Spain 2003, pp. 107–112
[22] Mayor G., Torrens J.: On a class of operators for expert systems. Internat. J. Intelligent Systems 8 (1993), No. 7, 771–778 DOI 10.1002/int.4550080703 | Zbl 0785.68087
[23] Pei D.: $R_0$ implication: characteristics and applications. Fuzzy Sets and Systems 131 (2002), 297–302 MR 1939842 | Zbl 1015.03034
[24] Trillas E., Campo, C. del, Cubillo S.: When QM-operators are implication functions and conditional fuzzy relations. Internat. J. Intelligent Systems 15 (2000), 647–655 DOI 10.1002/(SICI)1098-111X(200007)15:7<647::AID-INT5>3.0.CO;2-T | Zbl 0953.03031
Search
Partner of
