New results on overlap and grouping functions

On additive generators of overlap functions

“…We have also analyzed the influence of the migrativity, homogeneity and idempotency properties in the overlap functions obtained by such distortion and their respective additive generator pairs. Future theoretical work is concerned with the study of the concept of additive generator pair of a grouping function [2][3][4], extending the introductory work presented in [28], analyzing how some of the properties that can be demanded to a grouping function can be expressed in terms of its generator pair. We also intend to extend the concepts of additive generator pairs of overlap and grouping functions to the interval-valued setting, following the approach adopted in [29][30][31][32][33][34].…”

“…Some properties of overlap functions (e.g., migrativity, homogeneity, idempotency, convex combination) were also studied by Bedregal et al [4] and Jurio et al [2]. …”

“…4 Notice that the concept of additive generator pair of overlap functions introduced here is inspired by Viceník's work [12][13][14][15] related to additive generators of (non-continuous) t-norms, based on a pair of functions composed by a (non-continuous) one-place function and its pseudo-inverse. In this paper, since overlap functions are continuous, their additive generator pair is composed by continuous functions.…”

“…Finally, several studies have also been presented in recent years from a theoretical point of view, analyzing properties such as migrativity, homogeneity, idempotency or Lipschitzianity (see, e.g., the works by Bustince et al [1][2][3][4][5][6][7]). …”

Upper bounding overlaps by groupings

“…Thus researching theoretical properties of overlaps and groupings taking as references properties of t -norms and t -conorms looks to have sense. Actually, many results in the literature about overlaps and groupings are inspired in such a way [3,4]. Anyway, there are also some properties holding between t-norms and t -conorms that do not hold between overlaps and groupings.…”