Analyzing Properties of Fuzzy Implications Obtained via the Interval Constructor

This work considers an interval extension of fuzzy implications based on the best interval representation of usual fuzzy implications. The related properties of fuzzy implications can be naturally extended and the interval representation meets the optimality property and preserves the behaviors of the implications in the interval endpoints. Our discussion mainly focuses on the best interval representation 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.