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On two types of discrete implications. (English) Zbl 1084.03021
Summary: This paper deals with two kinds of implications defined from t-norms, t-conorms and strong negations on a finite chain \(L\): those defined through the expressions \(I(x, y) = S(N(x), T(x, y))\) and \(I(x, y) = S(T(N(x), N(y)), y)\). They are called QL-implications and NQL-implications respectively. We mainly study those QL- and NQL-implications derived from smooth t-norms and smooth t-conorms. It is characterized when functions defined in these ways are implication functions, and their analytical expressions are given. It is proved that both kinds of implications coincide. Some additional properties are studied like contrapositive symmetry, the exchange principle and others. In particular, it is proved that contrapositive symmetry holds if and only if \(S\) is the only Archimedean t-conorm on \(L,\) and \(T\) jointly with its \(N\)-dual t-conorm satisfy the Frank equation. Finally, some QL- and NQL-implications are also derived from non-smooth t-norms or non-smooth t-conorms and many examples are given showing that in this non-smooth case, QL- and NQL-implications remain strongly connected.

03B52 Fuzzy logic; logic of vagueness
68T37 Reasoning under uncertainty in the context of artificial intelligence
Full Text: DOI
[1] De Baets, B.; Mesiar, R., Triangular norms on product lattices, Fuzzy sets and systems, 104, 61-75, (1999) · Zbl 0935.03060
[2] Bustince, H.; Burillo, P.; Soria, F., Automorphisms, negations and implication operators, Fuzzy sets and systems, 134, 209-229, (2003) · Zbl 1010.03017
[3] Fodor, J.C., On fuzzy implication operators, Fuzzy sets and systems, 42, 293-300, (1991) · Zbl 0736.03006
[4] Fodor, J.C., Contrapositive symmetry on fuzzy implications, Fuzzy sets and systems, 69, 141-156, (1995) · Zbl 0845.03007
[5] Fodor, J.C., Smooth associative operations on finite ordinal scales, IEEE transactions on fuzzy systems, 8, 791-795, (2000)
[6] Ll. Godo, C. Sierra, A new approach to connective generation in the framework of expert systems using fuzzy logic, in: Proceedings of the XVIIIth ISMVL, Palma de Mallorca, 1988, pp. 157-162.
[7] Godo, L.; Torra, V., On aggregation operators for ordinal qualitative information, IEEE transactions on fuzzy systems, 8, 2, 143-154, (2000)
[8] Klir, G.J.; Yuan, B., Fuzzy sets and fuzzy logic. theory and applications, (1995), Prentice Hall New Jersey · Zbl 0915.03001
[9] Mas, M.; Mayor, G.; Torrens, J., t-operators and uninorms on a finite totally ordered set, International journal of intelligent systems, 14, 9, 909-922, (1999) · Zbl 0948.68173
[10] M. Mas, M. Monserrat, J. Torrens, QL-implications on a finite chain, in: Proceedings Eusflat-2003, Zittau, Germany, 2003, pp. 281-284.
[11] Mas, M.; Monserrat, M.; Torrens, J., S-implications and R-implications on a finite chain, Kybernetika, 40, 3-20, (2004) · Zbl 1249.94094
[12] Mas, M.; Monserrat, M.; Torrens, J., On left and right uninorms on a finite chain, Fuzzy sets and systems, 146, 3-17, (2004) · Zbl 1045.03029
[13] Mayor, G.; Torrens, J., On a class of operators for expert systems, International journal of intelligent systems, 8, 7, 771-778, (1993) · Zbl 0785.68087
[14] Mayor, G.; Torrens, J., Triangular norms in discrete settings, (), 189-230 · Zbl 1079.03012
[15] Pei, D., R_{0} implication: characteristics and applications, Fuzzy sets and systems, 131, 297-302, (2002) · Zbl 1015.03034
[16] Trillas, E.; del Campo, C.; Cubillo, S., When QM-operators are implication functions and conditional fuzzy relations, International journal of intelligent systems, 15, 647-655, (2000) · Zbl 0953.03031
[17] Trillas, E.; Alsina, C.; Renedo, E.; Pradera, A., On contra-symmetry and MPT conditionality in fuzzy logic, International journal of intelligent systems, 20, 313-326, (2005) · Zbl 1088.03025
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