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)), where S₁ and S₂ are two continuous t -conorms given as ordinal sums, and 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 S₂ in the interval [I(1,0),I(1,1)], we get its continuous solutions directly. If there are summands of S₂ in 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. 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 (0,1] × [0,1].

I. Introduction

During the past 30 years, fuzzy systems have been successfully used in many fields, such as in control systems [1]–[4], decision making, and signal processing. It is well known that fuzzy systems can approximate any continuous function to any desired accuracy (see [5]–[7]). However, a large rule base is required usually to achieve high accuracy. In many situations, the problem of rule explosion has become a major drawback which hinders the successful application of fuzzy systems.