On interval additive generators of interval overlap functions and interval grouping functions

Overlap and grouping functions have been proposed by Bustince et al. for applications in image processing, classification problems and decision making based on fuzzy preference relations. In this paper, firstly, we generalize the concepts of overlap and grouping functions to the notions of interval overlap functions and interval grouping functions and investigate some vital related properties. In particular, we give the construction methods to obtain interval overlap functions and interval grouping functions, which are the best interval representation of the overlap and grouping functions, respectively. And then, we introduce the concepts of interval additive generators of interval overlap functions and interval grouping functions, which offer convenience for the selection of proper interval overlap functions or interval grouping functions in concrete problems. We also prove that the best interval representation of the additive generator pair of an additively generated overlap function is an interval additive generator pair of the best interval representation of this overlap function and the case for grouping function is analogous.