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 $I(x,S_1(y,z))=S_2(I(x,y),I(x,z))$, when $S_1,S_2$ 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 $S_1,S_2$ are either both strict or nilpotent t-conorms. Using the obtained characterizations, we show that the previous equation does not hold when $S_1,S_2$ 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 $S_1,S_2$ 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 $S_1 = S_2$ are $\Phi$-conjugate to the Łukasiewicz t-conorm for some increasing bijection $\varphi$ of the unit interval, which is also a multiplicative generator of $T$.