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QL-implications versus D-implications. (English) Zbl 1249.03026
Summary: 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 paid to the relation between both kinds of implications. In the continuous case, the study of these implications is focused on 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.

03B52 Fuzzy logic; logic of vagueness
39B05 General theory of functional equations and inequalities
94D05 Fuzzy sets and logic (in connection with information, communication, or circuits theory)
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[1] Alsina C., Trillas E.: On the functional equation \(S_1(x,y)=S_2(x,T(N(x),y))\). Functional Equations, Results and Advances (Z. Daróczy and Z. Páles, Kluwer Academic Publishers, Dordrecht 2002, pp. 323-334 · Zbl 0996.39021
[2] Bustince H., Burillo, P., Soria F.: Automorphisms, negations and implication operators. Fuzzy Sets and Systems 134 (2003), 209-229 · Zbl 1010.03017
[3] Baets B. De: Model implicators and their characterization. Proc. First ICSC International Symposium on Fuzzy Logic (N. Steele, ICSC Academic Press, Zürich 1995, pp. A42-A49
[4] Fodor J. C.: On fuzzy implication operators. Fuzzy Sets and Systems 42 (1991), 293-300 · Zbl 0736.03006
[5] Fodor J. C.: Contrapositive symmetry on fuzzy implications. Fuzzy Sets and Systems 69 (1995), 141-156 · Zbl 0845.03007
[6] Frank M. J.: On the simultaneous associativity of \(F(x,y)\) and \(x + y - F(x,y)\). Aequationes Math. 19 (1979), 194-226 · 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 · Zbl 0941.03059
[8] Klement E. P., Mesiar, R., Pap E.: Triangular Norms. Kluwer Academic Publishers, Dordrecht 2000 · Zbl 1087.20041
[9] Mas M., Monserrat, M., Torrens J.: QL-Implications on a finite chain. 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 · Zbl 1084.03021
[11] Nachtegael M., Kerre E.: Classical and fuzzy approaches towards mathematical morphology. Fuzzy Techniques in Image Processing (E. Kerre and M. Nachtegael, 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 · 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 <a href=”http://dx.doi.org/10.1002/(SICI)1098-111X(200007)15:73.0.CO;2-T” target=”_blank”>DOI 10.1002/(SICI)1098-111X(200007)15:73.0.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 · Zbl 1088.03025
[15] Yager R. R.: Uninorms in fuzzy systems modelling. Fuzzy Sets and Systems 122 (2001), 167-175 · Zbl 0978.93007
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