Previous |  Up |  Next


t-norm; T-conorm; implication operator; QL-implication; D-implication
This paper deals with two kinds of fuzzy implications: QL and Dishkant implications. That is, those defined through the expressions $I(x,y) = S(N(x),T(x,y))$ and $I(x,y) = S(T(N(x),N(y)),y)$ respectively, where $T$ is a t-norm, $S$ is a t-conorm and $N$ is a strong negation. Special attention is due to the relation between both kinds of implications. In the continuous case, the study of these implications is focused in some of their properties (mainly the contrapositive symmetry and the exchange principle). Finally, the case of non continuous t-norms or non continuous t-conorms is studied, deriving new implications of both kinds and showing that they remain strongly connected.
[1] Alsina C., Trillas E.: On the functional equation $S_1(x,y)=S_2(x,T(N(x),y))$. In: Functional Equations, Results and Advances (Z. Daróczy and Z. Páles, eds.), Kluwer Academic Publishers, Dordrecht 2002, pp. 323–334 MR 1912725 | Zbl 0996.39021
[2] 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
[3] Baets B. De: Model implicators and their characterization. In: Proc. First ICSC International Symposium on Fuzzy Logic (N. Steele, ed.), ICSC Academic Press, Zürich 1995, pp. A42–A49
[4] 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
[5] Fodor J. C.: Contrapositive symmetry on fuzzy implications. Fuzzy Sets and Systems 69 (1995), 141–156 DOI 10.1016/0165-0114(94)00210-X | MR 1317882
[6] Frank M. J.: On the simultaneous associativity of $F(x,y)$ and $x + y - F(x,y)$. Aequationes Math. 19 (1979), 194–226 DOI 10.1007/BF02189866 | MR 0556722 | Zbl 0444.39003
[7] 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
[8] Klement E. P., Mesiar, R., Pap E.: Triangular Norms. Kluwer Academic Publishers, Dordrecht 2000 MR 1790096 | Zbl 1087.20041
[9] Mas M., Monserrat, M., Torrens J.: QL-Implications on a finite chain. In: Proc. Eusflat-2003, Zittau 2003, pp. 281–284
[10] Mas M., Monserrat, M., Torrens J.: On two types of discrete implications. Internat. J. Approx. Reason. 40 (2005), 262–279 DOI 10.1016/j.ijar.2005.05.001 | MR 2193766 | Zbl 1084.03021
[11] Nachtegael M., Kerre E.: Classical and fuzzy approaches towards mathematical morphology. In: Fuzzy Techniques in Image Processing (E. Kerre and M. Nachtegael, eds., Studies in Fuzziness and Soft Computing, Vol. 52), Physica–Verlag, Heidelberg 2000, pp. 3–57
[12] Pei D.: $R_0$ implication: characteristics and applications. Fuzzy Sets and Systems 131 (2002), 297–302 MR 1939842 | Zbl 1015.03034
[13] 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
[14] Trillas E., Alsina C., Renedo, E., Pradera A.: On contra-symmetry and MPT conditionality in fuzzy logic. Internat. J. Intelligent Systems 20 (2005), 313–326 DOI 10.1002/int.20068 | Zbl 1088.03025
[15] Yager R. R.: Uninorms in fuzzy systems modelling. Fuzzy Sets and Systems 122 (2001), 167–175 DOI 10.1016/S0165-0114(00)00027-0 | MR 1839955
Partner of
EuDML logo