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Two types of implications derived from uninorms. (English) Zbl 1125.03018
A uninorm is a function $$U: [0, 1]\times [0, 1]\to [0, 1]$$ which is associative, commutative, increasing in both variables and such that there exists an element $$e\in [0, 1]$$, called the neutral element, such that $$U(e, x)= x$$ for all $$x\in [0, 1]$$. $$U$$ becomes a t-norm if $$e=1$$ and a t-conorm if $$e=0$$. Thus, the theory of uninorms includes the theory of t-norms and t-conorms. A uninorm is conjunctive if $$U(1, 0)=0$$ and disjunctive if $$U(1, 0)=1$$.
Thus, a natural question is raised, how implications can be introduced on the basis of uninorms. There are essentially two ways: QL-implications $$I_Q(x, y)= U'(N(x), U(x, y))$$, where $$U$$ is a conjunctive and $$U'$$ a disjunctive uninorm and $$N$$ is a strong negation. The second implication is D-implication defined by $$I_D(x, y)= U'(U(N(x), N(y)), y)$$. The paper contains several technical results concerning the behavior of both kinds of implications in the continuous as well as in the non-continuous case, including necessary and necessary and sufficient conditions for their existence.

##### MSC:
 03B52 Fuzzy logic; logic of vagueness 68T37 Reasoning under uncertainty in the context of artificial intelligence
##### Keywords:
uninorm; QL-implication; D-implication; contraposition
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##### References:
 [1] Combs, W.E.; Andrews, J.E., Combinatorial rule explosion eliminated by a fuzzy rule configuration, IEEE trans. fuzzy syst., 6, 1-11, (1998) [2] De Baets, B., Idempotent uninorms, European J. oper. res., 118, 632-642, (1999) · Zbl 0933.03071 [3] De Baets, B.; Fodor, J., Residual operators of uninorms, Soft. comput., 3, 89-100, (1999) [4] Fodor, J.; Roubens, M., Fuzzy preference modelling and multicriteria decision support, theory and decision library, series D: system theory, knowledge engineering and problem solving, (1994), Kluwer Academic Publishers Dordrecht [5] Fodor, J.C.; Yager, R.R.; Rybalov, A., Structure of uninorms, Internat. J. uncertainty fuzziness knowledge-based syst., 5, 4, 411-427, (1997) · Zbl 1232.03015 [6] González, M.; Ruiz, D.; Torrens, J., Algebraic properties of fuzzy morphological operators based on uninorms, (), 27-38 [7] Hu, S.; Li, Z., The structure of continuous uni-norms, Fuzzy sets and systems, 124, 43-52, (2001) · Zbl 0989.03058 [8] Jenei, S., New family of triangular norms via contrapositive symmetrization of residuated implications, Fuzzy sets and systems, 110, 157-174, (2000) · Zbl 0941.03059 [9] Klement, E.P.; Mesiar, R.; Pap, E., Triangular norms, trends in logic, Vol. 18, (2000), Kluwer Academic Publishers Dordrecht [10] Martín, J.; Mayor, G.; Torrens, J., On locally internal monotonic operations, Fuzzy sets and systems, 137, 1, 27-42, (2003) · Zbl 1022.03038 [11] Mas, M.; Monserrat, M.; Torrens, J., QL-implications versus D-implications, Kybernetika, 42, 3, 351-366, (2006) · Zbl 1249.03026 [12] Ruiz, D.; Torrens, J., Residual implications and co-implications from idempotent uninorms, Kybernetika, 40, 21-38, (2004) · Zbl 1249.94095 [13] Ruiz-Aguilera, D.; Torrens, J., Distributivity of strong implications over conjunctive and disjunctive uninorms, Kybernetika, 42, 319-336, (2006) · Zbl 1249.03030 [14] Ruiz-Aguilera, D.; Torrens, J., Distributivity of residual implications over conjunctive and disjunctive uninorms, Fuzzy sets and systems, 158, 1, 23-37, (2007) · Zbl 1114.03022 [15] Trillas, E.; Alsina, C., On the law $$[p \wedge q \rightarrow r] \equiv [(p \rightarrow r) \vee(q \rightarrow r)]$$ in fuzzy logic, IEEE trans. fuzzy syst., 10, 84-88, (2002) [16] Trillas, E.; Alsina, C.; Renedo, E.; Pradera, A., On contra-symmetry and MPT conditionality in fuzzy logic, Internat. J. intell. syst., 20, 313-326, (2005) · Zbl 1088.03025 [17] Trillas, E.; del Campo, C.; Cubillo, S., When QM-operators are implication functions and conditional fuzzy relations, Internat. J. intell. syst., 15, 647-655, (2000) · Zbl 0953.03031 [18] E. Trillas, M. Mas, M. Monserrat, J. Torrens, On the representation of fuzzy if then rules, Internat. J. Approx. Reason., submitted for publication. · Zbl 1189.03033 [19] Yager, R.R., On some new classes of implication operators and their role in approximate reasoning, Inform. sci., 167, 193-216, (2004) · Zbl 1095.68119
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