This work considers an interval extension of fuzzy impli- cations based on the best interval representation of usual fuzzy implications. The related properties of fuzzy im- plications can be naturally extended and the interval re- presentation meets the optimality property and preserves the behaviors of the implications in the interval endpoints. Our discussion mainly focuses on the best interval repre- sentation of three important classes of fuzzy implications: S-implications, R-implications and QL-implications. We analyze sufficient and necessary conditions for these three classes of implications as inclusion-monotonic functions in both arguments satisfying the minimal properties of fuzzy implications.
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