Abstract
In this paper, we summarize the sufficient and necessary conditions of solutions for the distributive equation of implication I(x,T 1(y,z)) = T 2(I(x,y),I(x,z)) and characterize all solutions of the functional equations consisting of I(x,T 1(y,z)) = T 2(I(x,y),I(x,z)) and I(x,y) = I(N(y),N(x)), when T 1 is a continuous but not Archimedean triangular norm, T 2 is a continuous and Archimedean triangular norm, I is an unknown function, N is a strong negation. We also underline that our method can apply to the three other functional equations closely related to the above-mentioned functional equations.
This work is supported by National Natural Science Foundation of China (Nos. 60904041, 61165014) and Jiangxi Natural Science Foundation (No.2009GQS0055).
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Qin, F., Lu, M. (2011). On Distributive Equations of Implications and Contrapositive Symmetry Equations of Implications Based on a Continuous t-Norm. In: Tang, Y., Huynh, VN., Lawry, J. (eds) Integrated Uncertainty in Knowledge Modelling and Decision Making. IUKM 2011. Lecture Notes in Computer Science(), vol 7027. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-24918-1_14
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