Abstract
The law of importation between fuzzy implications and conjunctions is an important property in both the theory and the application of fuzzy logic. Although many interesting results have been reported in the literature, the related open problem of finding all possible pairs of implications and conjunctions satisfying the law of importation has not been fully solved. As a continuation of two articles S. Massanet et al. (2018) [23]; S. Massanet et al. (2018) [24], this article aims to characterize fuzzy implication solutions with a continuous natural negation to the law of importation with respect to a fixed conjunctive uninorm having continuous underlying operators in several cases. Such solutions for the cases where the continuous underlying triangular norm and triangular conorm of a prefixed uninorm are idempotent or Archimedean are completely characterized, and the solutions for the cases where the continuous underlying operators are given by ordinal sums are characterized under some additional assumptions. Finally, some comments on relations to recent independent work W.-H. Li and F. Qin (2021) [25] are given. These two independent articles will go a further step towards answering the open problem regarding the law of importation.
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