# Article

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Keywords:
$t$-norm; $t$-conorm; uninorm; implication operator; S-implication; R-implication; distributivity
Summary:
This paper deals with implications defined from disjunctive uninorms $U$ by the expression $I(x,y)=U(N(x),y)$ where $N$ is a strong negation. The main goal is to solve the functional equation derived from the distributivity condition of these implications over conjunctive and disjunctive uninorms. Special cases are considered when the conjunctive and disjunctive uninorm are a $t$-norm or a $t$-conorm respectively. The obtained results show a lot of new solutions generalyzing those obtained in previous works when the implications are derived from $t$-conorms.
References:
[1] Balasubramaniam J., Rao C. J. M.: On the distributivity of implication operators over T and S norms. IEEE Trans. Fuzzy Systems 12 (2004), 194–198 DOI 10.1109/TFUZZ.2004.825075
[2] Combs W. E.: Combinatorial rule explosion eliminated by a fuzzy rule configuration. IEEE Trans. Fuzzy Systems 6 (1998), 1–11 DOI 10.1109/91.660804
[3] Combs W. E., Andrews J. E.: Author’s reply. IEEE Trans. Fuzzy Systems 7 (1999), 371 DOI 10.1109/TFUZZ.1999.771094
[4] Combs W. E., Andrews J. E.: Author’s reply. IEEE Trans. Fuzzy Systems 7 (1999), 478–479 DOI 10.1109/TFUZZ.1999.771094
[5] Baets B. De, Fodor J. C.: Residual operators of uninorms. Soft Computing 3 (1999), 89–100 DOI 10.1007/s005000050057
[6] Dick S., Kandel A.: Comments on “combinatorial rule explosion eliminated by a fuzzy rule configuration”. IEEE Trans. Fuzzy Systems 7 (1999), 475–477 DOI 10.1109/91.784213
[7] Fodor J. C., Yager R. R., Rybalov A.: Structure of uninorms. Internat. J. Uncertainty, Fuzziness and Knowledge-based Systems 5 (1997), 4, 411–427 DOI 10.1142/S0218488597000312 | MR 1471619 | Zbl 1232.03015
[8] González M., Ruiz, D., Torrens J.: Algebraic properties of fuzzy morphological operators based on uninorms. In: Artificial Intelligence Research and Development (I. Aguiló, L. Valverde, and M. Escrig, eds.), IOS Press 2003, pp. 27–38
[9] Klement E. P., Mesiar, R., Pap E.: Triangular Norms. Kluwer Academic Publishers, Dordrecht 2000 MR 1790096 | Zbl 1087.20041
[10] Martín J., Mayor, G., Torrens J.: On locally internal monotonic operations. Fuzzy Sets and Systems 137 (2003), 1, 27–42 DOI 10.1016/S0165-0114(02)00430-X | MR 1992696 | Zbl 1022.03038
[11] Mas M., Mayor, G., Torrens J.: The distributivity condition for uninorms and $t$-operators. Fuzzy Sets and Systems 128 (2002), 209–225 MR 1908427 | Zbl 1005.03047
[12] Mas M., Mayor, G., Torrens J.: Corrigendum to “The distributivity condition for uninorms and $t$-operators”. Fuzzy Sets and Systems 128 (2002), 209–225, Fuzzy Sets and Systems 153 (2005), 297–299 MR 1908427
[13] Mendel J. M., Liang Q.: Comments on “combinatorial rule explosion eliminated by a fuzzy rule configuration”. IEEE Trans. Fuzzy Systems 7 (1999), 369–371 DOI 10.1109/91.771093
[14] Ruiz D., Torrens J.: Distributive idempotent uninorms. Internat. J. Uncertainty, Fuzziness and Knowledge-Based Systems 11 (2003), 413–428 DOI 10.1142/S0218488503002168 | MR 2007849 | Zbl 1074.03026
[15] Ruiz D., Torrens J.: Residual implications and co-implications from idempotent uninorms. Kybernetika 40 (2004), 21–38 MR 2068596
[16] Ruiz D., Torrens J.: Distributivity and conditional distributivity of a uninorm and a continuous $t$-conorm. IEEE Trans. Fuzzy Systems 14 (2006), 180–190 DOI 10.1109/TFUZZ.2005.864087
[17] Ruiz D., Torrens J.: Distributive residual implications from uninorms. In: Proc. EUSFLAT-2005, Barcelona 2005, pp. 369–374
[18] 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 Systems 10 (2002), 84–88

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