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 $I(x,S_1(y,z))=S_2(I(x,y),I(x,z))$ , where $S_1$ and $S_2$ are two continuous t -conorms given as ordinal sums, and $I:[\hbox {0},{1}]^2\rightarrow [\hbox {0},{1}]$ is a binary function which is increasing with respect to the second place. If there is no summand of $S_2$ in the interval $[I({1},\hbox {0}),I({1},{1})]$ , we get its continuous solutions directly. If there are summands of $S_2$ in the interval $[I({1},\hbox {0}),I({1},{1})]$ , 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. When $I$ is 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 $(\hbox {0},{1}]\times [\hbox {0},{1}]$ .