Distributive Equations of Fuzzy Implications Based on Continuous Triangular Conorms Given as Ordinal Sums

Recently, the distributive equations of fuzzy implications based on t -norms or t-conorms have become a focus of research. The solutions to these equations can help people design the structures of fuzzy systems in such a way that the number of rules is largely reduced. This paper studies the distributive functional equation I(x,S1(y,z))=S2(I(x,y),I(x,z)), whereS1andS2are two continuous t -conorms given as ordinal sums, andI:[0,1]2→ [0,1] is a binary function which is increasing with respect to the second place. If there is no summand ofS2in the interval [I(1,0),I(1,1)], we get its continuous solutions directly. If there are summands ofS2in the interval [I(1,0),I(1,1)], by defining a new concept called feasible correspondence and using this concept, we describe the solvability of the distributive equation above and characterize its general continuous solutions. WhenIis restricted to fuzzy implications, it is showed that there is no continuous solution to this equation. We characterize its fuzzy implication solutions, which are continuous on (0,1] × [0,1].