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(<i>y</i>,<i>z</i>))=<i>S</i>₂(<i>I</i>(<i>x</i>,<i>y</i>),<i>I</i>(<i>x</i>,<i>z</i>)), where <i>S</i>₁ and <i>S</i>₂ are two continuous t -conorms given as ordinal sums, and <i>I</i>:[0,1]²→ [0,1] is a binary function which is increasing with respect to the second place. If there is no summand of <i>S</i>₂ in the interval [<i>I</i>(1,0),<i>I</i>(1,1)], we get its continuous solutions directly. If there are summands of <i>S</i>₂ in the interval [<i>I</i>(1,0),<i>I</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>I</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 (0,1] × [0,1].