On the Distributivity of Fuzzy Implications Over Nilpotent or Strict Triangular Conorms
Michał Baczyński, Balasubramaniam Jayaram · IEEE Transactions on Fuzzy Systems · 2009
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$T$.