On interval (G,N)-implications and (O,G,N)-implications derived from interval overlap and grouping functions

As special sorts of nonassociative binary aggregation operators, overlap and grouping functions show tremendous potential in dealing with classification problems and making decisions due to fuzzy preference relations. In this paper, we introduce two kinds of interval extensions for (G,N)-implications and QL-implications derived from overlap and grouping functions, namely interval (G,N)-implications and interval (O,G,N)-implications, respectively. Firstly, we investigate some desirable properties of interval (G,N)-implications and its relationships with interval automorphisms. Particularly, using the notion of the best interval representation, we provide a characterization of interval (G,N)-implications. And then, we give a necessary and sufficient condition and an only sufficient and not necessary condition for the interval (O,G,N)-operation to be an interval (O,G,N)-implication. Meanwhile, we analyze various main properties of interval (O,G,N)-operations, in particular, of interval (O,G,N)-implications, illustrating the relationships between interval automorphisms and them. We also obtain interesting conclusions of interval (O,G,N)-implications, such as its convex sum and so on. Finally, we study the intersections between interval (G,N)-implications and interval (O,G,N)-implications.