A function that takes n numbers as input and outputs one number is said to be homogeneous whenever the result of multiplying each input by a certain factor λ yields the original output multiplied by that same factor.This concept has been extended by the notion of abstract homogeneity, which generalizes the product in the expression of homogeneity by a general function g and the effect of the factor λ by an automorphism.However, the effect of parameter λ remains unchanged for all the input values.In this study, we generalize further the condition of abstract homogeneity by introducing ℱ-homogeneity, which is defined with respect to a family of functions, enabling a different behavior for each of the inputs.Next, we study the properties that are satisfied by this family of functions and, moreover, we link this concept with the condition of directional monotonicity, which is a trendy property in the framework of aggregation functions.To achieve that, we generalize directional monotonicity by ℱ directional monotonicity, which is defined with respect to a family of functions ℱ and a family of vectors  .Finally, we show how the introduced concepts could be applied

A new concept of equivalence between intervals and the induced indistinguishability between interval-valued (IV) fuzzy sets are proposed and considered. A new notion of the degree of IV equivalence is presented where partial or linear orders and the width of intervals are involved reflecting uncertainty. Furthermore, construction methods of the considered equivalences are provided and the relation between them and other notions of IV equivalences are studied. Finally, a methodology to apply the proposed equivalences in the field of image processing is shown, along with an illustrative example.

Imbalanced classification problems are attracting the attention of the research community because they are prevalent in real-world problems and they impose extra difficulties for learning methods. Fuzzy rule-based classification systems have been applied to cope with these problems, mostly together with sampling techniques. In this paper, we define a new fuzzy association rule-based classifier, named FARCI, to tackle directly imbalanced classification problems. Our new proposal belongs to the algorithm modification category, since it is constructed on the basis of the state-of-the-art fuzzy classifier FARC–HD. Specifically, we modify its three learning stages, aiming at boosting the number of fuzzy rules of the minority class as well as simplifying them and, for the sake of handling unequal fuzzy rule lengths, we also change the matching degree computation, which is a key step of the inference process and it is also involved in the learning process. In the experimental study, we analyze the effectiveness of each one of the new components in terms of performance, F-score, and rule base size. Moreover, we also show the superiority of the new method when compared versus FARC–HD alongside sampling techniques, another algorithm modification approach, two cost-sensitive methods and an ensemble.