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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)
Xie, A. ; School of Mathematics, Shandong University, Jinan, China ; Li, C. ; Liu, H.

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 $I:[hbox{0},{1}]^2rightarrow [hbox{0},{1}]$ 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:

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

Date of Publication:

June 2013

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