Some Advances on the Solution of the Isabel AGUIL ´ O a,b,1 , Sebastia MASSANET a,b and Juan Vicente RIERA a,b a SCOPIA research group, Dept. of Mathematics and Computer Science, University of the Balearic Islands, 07122 Palma, Spain b Health Research Institute of the Balearic Islands (IdISBa), 07010 Palma, Spain Abstract. The law of importation has attracted the interest of many researchers de- voted to fuzzy implication functions in the last decades. This property has several important applications, especially in approximate reasoning and image processing. Several generalizations of this property have been proposed. Specifically, one gen- eralization related to the law of migrativity was recently introduced by Baczy´ nski et al. in which two fuzzy implication functions are involved. In this paper, some advances on the solution of this functional equation for the particular case where the involved fuzzy conjunction is a t-norm are presented. Indeed, a complete char- acterization of all those pairs of fuzzy implication functions with a strict natural fuzzy negation satisfying the generalized law of importation is achieved. Keywords. Fuzzy implication function, T-norm, Functional equation, Generalized law of importation, Natural negation 1. Introduction One of the main lines of research in fuzzy logic is the theoretical study of fuzzy impli- cation functions. These operators, which are the generalization of the classical implica- tion to the fuzzy framework, have important applications in approximate reasoning or image processing [2]. For a concrete application, fuzzy implication functions fulfilling some additional algebraical properties are needed and therefore, the characterization of such operators (obtained by solving functional equations) provides a bunch of operators useful for those applications. One of such additional properties is the law of importation with respect to a fuzzy conjunction C. This property is given by I (C(x, y), z)= I (x, I (y, z)) for all x, y, z ∈ [0, 1], (LI) (1) where I is a fuzzy implication function and C a fuzzy conjunction (for more details see [1,6]). The significance of the law of importation is related to the simplification of the 1 Corresponding Author: Isabel Aguil´ o, Dept. of Mathematics and Computer Science, University of the Balearic Islands, Ctra. de Valldemossa, Km.7.5, 07122 Palma, Spain; E-mail: isabel.aguilo@uib.es. Generalized Law of Importation for Fuzzy Implication Functions Artificial Intelligence Research and Development M. Villaret et al. (Eds.) © 2021 The authors and IOS Press. This article is published online with Open Access by IOS Press and distributed under the terms of the Creative Commons Attribution Non-Commercial License 4.0 (CC BY-NC 4.0). doi:10.3233/FAIA210119 86