1063-6706 (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TFUZZ.2017.2756832, IEEE Transactions on Fuzzy Systems IEEE TRANSACTIONS ON FUZZY SYSTEMS 1 Characterization of a class of fuzzy implication functions satisfying the law of importation with respect to a fixed uninorm (Part I) Sebastia Massanet, Daniel Ruiz-Aguilera and Joan Torrens 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 which 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, the current one and the forthcoming paper [1]. In particular, in this paper the case of uninorms in Umin 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 forthcoming paper [1]. Index Terms—Fuzzy implication function, law of importation, uninorm, fuzzy negation, (U, N )-implication. I. I NTRODUCTION Fuzzy implication functions are the generalization of binary implications in classical logic to the framework of fuzzy logic and consequently they are used to perform fuzzy conditionals [2]–[4]. In addition of modeling fuzzy conditionals, they are also used to perform backward and forward inferences in any fuzzy rules based system through the inference rules of modus ponens and modus tollens [4]–[6]. Moreover, fuzzy implication functions have proven to be useful not only in fuzzy control and approximate reasoning, but also in many other fields like fuzzy relational equations [4], fuzzy DI- subsethood measures and image processing [7], [8], fuzzy morphological operators [9], computing with words [4], data mining [10] and rough sets [11], among others. Due to this great quantity of applications, fuzzy implication functions have attracted the efforts of many researchers also from the purely theoretical perspective. See for instance the surveys [4], [12] and [13] and the books [14] and [15], entirely devoted to fuzzy implication functions. In this framework, the analysis of some interesting properties of fuzzy implication functions is one of the most studied topics. In almost all the cases the interest of each property comes from its specific All the authors are with the Soft Computing, Image Processing and Aggregation research group (SCOPIA), Dept. Mathematics and Computer Science, University of the Balearic Islands, E-07122 Palma, Spain and the Balearic Islands Health Research Institute (IdISBa), 07010 Palma, Spain (email: {s.massanet,daniel.ruiz,jts224}@uib.es) applications and its study usually reduces to the solution of some functional equation. Among these additional properties, the law of importation with respect to a t-norm T or a conjunctive uninorm U has attracted the interest of many researchers. This property is extremely related to the exchange principle (see [16]), an almost indispensable property in all the fields where fuzzy implication functions are applied. Moreover, it has several important applications, specially in approximate reasoning and image processing. Namely, • it has proven to be useful in simplifying the process of applying the Compositional Rule of Inference (CRI) of Zadeh (see [17]) reducing its complexity through the so- called Hierarchical CRI, see [15] and [18], • in image processing, and in particular in fuzzy mathema- tical morphology, this property is required to the t-norm T (or the uninorm U ) and the fuzzy implication function I in order to obtain fuzzy morphological operators with desirable algebraical properties, see [9], [19] for more details. The law of importation is defined as I (T (x, y),z)= I (x, I (y,z)) for all x, y, z ∈ [0, 1], (LI) where T is a t-norm (a conjunctive uninorm, or a more general conjunction) and I is a fuzzy implication function. It has been studied in [15], [16], [18], [20], [21] and [22]. Moreover, the law of importation has also been used in new characterizations of some classes of implications like (S, N )-implications, R- implications, (U, N )-implications and RU -implications (see [16]), and also to characterize Yager’s implications (see [23]). In spite of the publication of all these works devoted to the law of importation, there are still some open problems invol- ving this property. Namely, given any conjunctive uninorm U (in particular U could be a t-norm T ), it is an open problem posed several years ago, to find all fuzzy implication functions I such that they satisfy the law of importation with respect to this fixed uninorm U (see Problem 8.1 in [24], [16] and Problem 12.2 in [13]). That is, to find all fuzzy implication functions such that I (U (x, y),z)= I (x, I (y,z)) for all x, y, z ∈ [0, 1], (LI U ) being U any fixed conjunctive uninorm. The particular case when U is a t-norm T was recently partially solved in [22] where all fuzzy implication functions having a continuous natural negation and satisfying the law