Interval-valued implications and interval-valued strong equality index with admissible orders

In this work we introduce the definition of interval-valued fuzzy implication function with respect to any total order between intervals. We also present different construction methods for such functions. We show that the advantage of our definitions and constructions lays on that we can adapt to the interval-valued case any inequality in the fuzzy setting, as the one of the generalized modus ponens. We also introduce a strong equality measure between interval-valued fuzzy sets, in which we take the width of the considered intervals into account, and, finally, we discuss a construction method for this measure using implication functions with respect to total orders.