Abstract
Overlap and grouping functions have been studied by many scholars since they were proposed because of their successful application in various practical problems such as in image processing, decision-making, classifications etc.. At the same time, 0-overlap functions and 1-grouping functions as their generalization are not only necessary to characterize interval overlap and grouping functions, but also the proposed generalized overlap functions based on 0-overlap functions have been successfully applied to classification problem. In this paper, we focus on the unification of 0-overlap functions and 1-grouping functions. Specifically, firstly, we introduce the concept of Θ−Ξ functions, which is the unification of 0-overlap functions and 1-grouping functions, and give some basic properties of Θ−Ξ functions. Secondly, we investigate the construction methods of Θ−Ξ functions on the basis of two classes of peculiar binary functions, so-called (e,k)-pseudo-automorphisms and (e,k)-aggregation functions with e,k belonging to unit closed interval, respectively. Finally, we obtain two equivalent characterizations of Θ−Ξ functions.
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