Open Access
General grouping functions
(Springer, 2020) Santos, Helida; Pereira Dimuro, Graçaliz; Da Cruz Asmus, Tiago; Sanz Delgado, José Antonio; Fernández Fernández, Francisco Javier; Bustince Sola, Humberto; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC; Estadística, Informática y Matemáticas
Some aggregation functions that are not necessarily associative, namely overlap and grouping functions, have called the attention of many researchers in the recent past. This is probably due to the fact that they are a richer class of operators whenever one compares with other classes of aggregation functions, such as t-norms and t-conorms, respectively. In the present work we introduce a more general proposal for disjunctive n-ary aggregation functions entitled general grouping functions, in order to be used in problems that admit n dimensional inputs in a more flexible manner, allowing their application in different contexts. We present some new interesting results, like the characterization of that operator and also provide different construction methods.
Open Access
On construction methods of (interval-valued) general grouping functions
(Springer, 2022) Pereira Dimuro, Graçaliz; Da Cruz Asmus, Tiago; Pinheiro, Jocivania; Santos, Helida; Borges, Eduardo N.; Lucca, Giancarlo; Rodríguez Martínez, Iosu; Mesiar, Radko; Bustince Sola, Humberto; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika
Recently, several theoretical and applied studies on grouping functions and overlap functions appeared in the literature, mainly because of their flexibility when comparing them with the popular aggregation operators t-conorms and t-norms, respectively. Additionally, they constitute richer classes of disjunction/conjunction operations than t-norms and t-conorms. In particular, grouping functions have been applied as the disjunction operator in several problems, like decision making based on fuzzy preference relations. In this case, when performing pairwise comparisons, grouping functions allow one to evaluate the measure of the amount of evidence in favor of either of two given alternatives. However, grouping functions are not associative. Then, in order to allow them to be applied in n-dimensional problems, such as the pooling layer of neural networks, some generalizations were introduced, namely, n-dimensional grouping functions and the more flexible general grouping functions, the latter for enlarging the scope of applications. Then, in order to h andle uncertainty on the definition of the membership functions in real-life problems, n-dimensional and general interval-valued grouping functions were proposed. This paper aims at providing new constructions methods of general (interval-valued) grouping functions, also providing some examples.
Open Access
Grouping indices: definition, properties and construction methods
(IEEE, 2024-08-05) Pereira Dimuro, Graçaliz; Santos, Helida; Urío Larrea, Asier; Da Cruz Asmus, Tiago; Lucca, Giancarlo; Camargo, Heloisa A.; Parodi, Maria Eugênia; Bustince Sola, Humberto; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC
Zadeh has defined its consistency index OZ in order to compare two fuzzy subsets A and B of a referential set U, in the sense that “The lower OZ (A, B) is, the closer to ∅ A ∩ B is.” Several axiomatizations of this concept can be found in the literature, in terms of overlap indices, allowing its application in different contexts, as in classification and clustering. The present work, in an opposite direction, introduces the grouping index, a comparison index G of two fuzzy subsets A and B that provides a measure of the proximity of the fuzzy union of A and B to the referential set U, that is, ‘the higher G(A, B) is, the closer to U A∪ B is.” Many important properties are analyzed, and construction methods are presented, whereas establishing the formal relation between overlap and grouping indices via duality. The theory is illustrated by several detailed examples.
Open Access
Constructing interval-valued fuzzy material implication functions derived from general interval-valued grouping functions
(IEEE, 2022) Pereira Dimuro, Graçaliz; Santos, Helida; Da Cruz Asmus, Tiago; Wieczynski, Jonata; Pinheiro, Jocivania; Bedregal, Benjamin; Bustince Sola, Humberto; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC
Grouping functions and their dual counterpart, overlap functions, have drawn the attention of many authors, mainly because they constitute a richer class of operators compared to other types of aggregation functions. Grouping functions are a useful theoretical tool to be applied in various problems, like decision making based on fuzzy preference relations. In pairwise comparisons, for instance, those functions allow one to convey the measure of the amount of evidence in favor of either of two given alternatives. Recently, some generalizations of grouping functions were proposed, such as (i) the n-dimensional grouping functions and the more flexible general grouping functions, which allowed their application in n-dimensional problems, and (ii) n-dimensional and general interval-valued grouping functions, in order to handle uncertainty on the definition of the membership functions in real-life problems. Taking into account the importance of interval-valued fuzzy implication functions in several application problems under uncertainty, such as fuzzy inference mechanisms, this paper aims at introducing a new class of interval-valued fuzzy material implication functions. We study their properties, characterizations, construction methods and provide examples.

Pereira Dimuro, Graçaliz

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