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 , T 1 ( y , z ) ) = T 2 ( I ( x , y ) , I ( x , z ) ) . Here we study 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 non-Archimedean triangular norm, T 2 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 T 1 and T 2 are continuous and Archimedean. Our methods are suitable for three other distributivity equations of implications closely related to those mentioned above.