On the Distributivity of Fuzzy Implications Over Nilpotent or Strict Triangular Conorms

Recently, many works have appeared in this very journal dealing with the distributivity of fuzzy implications over t-norms and t-conorms. These equations have a very important role to play in efficient inferencing in approximate reasoning, especially fuzzy control systems. Of all the four equations considered, the equation I(x, S1 (y,z)) = S2(I(x,y),I(x,z)), when S1,S2 are both t-conorms and I is an R-implication obtained from a strict t-norm, was not solved. In this paper, we characterize functions I that satisfy the previous functional equation when S1,S2 are either both strict or nilpotent t-conorms. Using the obtained characterizations, we show that the previous equation does not hold when S1,S2 are either both strict or nilpotent t-conorms, and I is a continuous fuzzy implication. Moreover, the previous equation does not hold when I is an R -implication obtained from a strict t-norm, and S1,S2 are both strict t-conorms, while it holds for an R-implication I obtained from a strict t-norm T if and only if the t-conorms S1 = S2 are Phi-conjugate to the Lukasiewicz t-conorm for some increasing bijection phi of the unit interval, which is also a multiplicative generator of T.