On the distributivity of fuzzy implications over continuous Archimedean t-conorms and continuous t-conorms given as ordinal sums. (English) Zbl 1279.03048
Authors’ abstract: In this paper, we investigate the distributive functional equation \(I(x,S_1 (y,z))=S_2 (I(x,y),I(x,z))\), where \(I:[0,1]^2 \to [0,1]\) is an unknown function, \(S_2\) a continuous Archimedean t-conorm and \(S_1\) a continuous t-conorm given as an ordinal sum. First, based on the special case with one summand in the ordinal sum of \(S_1\), all the sufficient and necessary conditions of solutions to the distributive equation above are given and the characterization of its continuous solutions is derived. It is shown that the distributive equation does not have continuous fuzzy implication solutions. Subsequently, we characterize its non-continuous fuzzy implication solutions. Finally, it is pointed out that the case with finite summands in the ordinal sum of \(S_1\) is equivalent to the one with one summand.
Reviewer: Michał Baczyński (Katowice)
MSC:
| 03B52 | Fuzzy logic; logic of vagueness |
| 03E72 | Theory of fuzzy sets, etc. |
| 26B99 | Functions of several variables |
| 39B99 | Functional equations and inequalities |
References:
| [1] | Aczél, J., Lectures on Functional Equations and their Applications (1966), Academic Press: Academic Press New York · Zbl 0139.09301 |
| [2] | Baczyński, M., On a class of distributive fuzzy implications, Int. J. Uncertainty Fuzziness Knowl.-Based Syst., 9, 229-238 (2001) · Zbl 1113.03315 |
| [3] | Baczyński, M., On the distributivity of fuzzy implications over continuous and Archimedean triangular conorms, Fuzzy Sets Syst., 161, 1406-1419 (2010) · Zbl 1204.03029 |
| [4] | Baczyński, M.; Jayaram, B., On the distributivity of fuzzy implications over nilpotent or strict triangular conorms, IEEE Trans. Fuzzy Syst., 17, 590-603 (2009) |
| [5] | Balasubramaniam, J.; Rao, C. J.M., On the distributivity of implication operators over \(T\)- and \(S\)-norms, IEEE Trans. Fuzzy Syst., 12, 194-198 (2004) |
| [6] | Combs, W. E.; Andrews, J. E., Combinatorial rule explosion eliminated by a fuzzy rule configuration, IEEE Trans. Fuzzy Syst., 6, 1-11 (1998) |
| [7] | Combs, W. E., Author’s reply, IEEE Trans. Fuzzy Syst., 7, 371-373 (1999) |
| [8] | Combs, W. E.; Andrews, J. E., Author’s reply, IEEE Trans. Fuzzy Syst., 7, 477-478 (1999) |
| [9] | Fodor, J.; Roubens, M., Fuzzy Preference Modelling and Multicriteria Decision Support (1994), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0827.90002 |
| [10] | Klement, E. P.; Mesiar, R.; Pap, E., Triangular Norms (2000), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0972.03002 |
| [11] | Ling, C. H., Representation of associative functions, Publ. Math. Debrecen, 12, 189-212 (1965) · Zbl 0137.26401 |
| [12] | Qin, F.; Yang, L., Distributive equations of implications based on nilpotent triangular norms, Int. J. Approximate Reasoning, 51, 984-992 (2010) · Zbl 1226.03036 |
| [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 Syst., 158, 23-37 (2007) · Zbl 1114.03022 |
| [15] | Ruiz, D.; Torrens, J., S- and R-implications from uninorms continuous in \((0,1)^2\) and their distributivity over uninorms, Fuzzy Sets Syst., 160, 832-852 (2009) · Zbl 1184.03014 |
| [16] | Trillas, E.; Alsina, C., On the law \([p \wedge q \to r] = [(p \to r) \vee(q \to r)]\) in fuzzy logic, IEEE Trans. Fuzzy Syst., 10, 84-88 (2002) |
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