A survey of the distributivity of implications over continuous T-norms and the simultaneous satisfaction of the contrapositive symmetry

Abstract

In this paper, we summarize the sufficient and necessary conditions of solutions for the distributivity equation of implication I(x,T 1(y,z)) = T 2(I(x,y),I(x,z)) and characterize all solutions of the system of 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 triangular norm, T 2 is a continuous Archimedean triangular norm, I is an unknown function and N is a strong negation. We also underline that our method can be applied to other distributivity functional equations closely related to the above mentioned distributivity equation. © Springer-Verlag Berlin Heidelberg 2013.