Characterization of Fuzzy Implication Functions With a Continuous Natural Negation Satisfying the Law of Importation With a Fixed t-Norm

The law of importation is an important property of fuzzy implication functions with interesting applications in approximate reasoning and image processing. This property has been extensively studied, and some open problems have been posed in the literature. In particular, in this paper, we partially solve an open problem related to this property posed some years ago. Specifically, given a fixed t-norm T, all fuzzy implication functions with continuous natural negation that satisfy the law of importation with this t-norm T are characterized. This characterization is especially detailed for the case of any continuous t-norm T, and particular cases are given for the minimum t-norm, for any continuous Archimedean t-norm, and for any ordinal sum of continuous Archimedean t-norms. For noncontinuous t-norms, the particular cases of the drastic t-norm and the nilpotent minimum t-norm are also presented separately. Finally, characterizations of some well-known fuzzy implication functions are also deduced from the presented results.