Revisiting a Functional Inequality on Fuzzy Implication Functions Related to the Inclusion of Fuzzy Sets.

In the last decades, fuzzy implication functions have proved their potential in many applications. In each of these applications, the fuzzy implication function must satisfy some additional properties whose study leads often to solving a functional equation or inequality [2,3]. This is the case of the functional inequality given by T(I(x, y), z) <= I(x, T(y, z)) where T is a t-norm and I a fuzzy implication function. This inequality appears in the setting of fuzzy object oriented models when modelling the degree of inclusion of a fuzzy set into another fuzzy set. In [8], this functional inequality was introduced and several results were proved when T is a continuous t-norm. In particular, the authors studied some particular cases of (S,N), R, QL and D-implications obtaining several solutions. Nevertheless, many cases of these families of fuzzy implication functions have not been studied yet as well as other well-known families of fuzzy implication functions. Following this line of research, in this paper, we deepen the study of this functional inequality. First, some general results are presented proving some necessary conditions on T and I to satisfy the inequality. After that, the results on (S,N) and R-implications are expanded and some solutions on the family of f -generated implications are presented. Finally, we study the preservation of this functional inequality for the most well-known construction methods of fuzzy implication functions. It is proved that some of them preserve this law and consequently, a plethora of new solutions is available through these methods.