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

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3 Author(s)
Aifang Xie ; School of Mathematics, Shandong University, Jinan, China ; Cheng Li ; Huawen Liu

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}] .

Published in:

IEEE Transactions on Fuzzy Systems  (Volume:21 ,  Issue: 3 )