Characterization of a Class of Fuzzy Implication Functions Satisfying the Law of Importation With Respect to a Fixed Uninorm—Part I

Abstract

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 proposed 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 uninorm, all fuzzy implication functions that satisfy the law of importation with respect to this uninorm, and having an α-section that is a continuous negation, are characterized. This characterization is specially detailed for the case of uninorms lying in each one of the most usual classes of uninorms. This is done in two different papers, this paper and the forthcoming paper (S. Massanet, D. Ruiz-Aguilera, and J. Torrens, “Characterization of a class of fuzzy implication functions satisfying the law of importation with a fixed uninorm-Part II,” <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">IEEE Trans. Fuzzy Syst.</i> , to be published). In particular, in this paper the case of uninorms in <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">U</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">min</sub> is solved, whereas the cases where the uninorm is in the other usual classes (that is, idempotent, representable, and continuous in the open unit square) are left for the above-mentioned paper.