On distributivity equations of implications and contrapositive symmetry equations of implications

To avoid combinatorial rule explosion in fuzzy reasoning, we recently obtained new solutions of the distributivity equation of implication I(x,T1(y,z))=T2(I(x,y),I(x,z))I(x,T1(y,z))=T2(I(x,y),I(x,z)). Here we study and characterize all solutions of the functional equations consisting of I(x,T1(y,z))=T2(I(x,y),I(x,z))I(x,T1(y,z))=T2(I(x,y),I(x,z)) and I(x,y)=I(N(y),N(x))I(x,y)=I(N(y),N(x)) when T1T1 is a continuous but non-Archimedean triangular norm, T2T2 is a continuous and Archimedean triangular norm, I is an unknown function, and N   is a strong negation. It should be noted that these results differ from the ones obtained by Qin and Yang when both T1T1 and T2T2 are continuous and Archimedean. Our methods are suitable for three other distributivity equations of implications closely related to those mentioned above.