On distributive equations of implications and contrapositive symmetry equations of implications based on a continuous t-Norm

Authors:
Feng Qin;Meihua Lu
Affiliations:
College of Mathematics and Information Science, Nanchang Hangkong University and College of Mathematics and Information Science, Jiangxi Normal University, Nanchang, P.R. China;College of Mathematics and Information Science, Jiangxi Normal University, Nanchang, P.R. China
Venue:
IUKM'11 Proceedings of the 2011 international conference on Integrated uncertainty in knowledge modelling and decision making
Year:
2011

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Abstract

In this paper, we summarize the sufficient and necessary conditions of solutions for the distributive equation of implication I(x, T1(y, z)) = T2(I(x, y), I(x, z)) and characterize all solutions of the functional equations consisting of I(x, T1(y, z)) = T2(I(x, y), I(x, z)) and I(x, y) = I(N(y),N(x)), when T1 is a continuous but not Archimedean triangular norm, T2 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.