On possibilistic representations of fuzzy intervals

It is acknowledged that a fuzzy interval has two equivalent representations given in terms of the so called Left and Right sides of the membership function (LR-representation) or in terms of the Lower and Upper branches defining the endpoints of the α-cuts (LU-representation). In this paper we suggest an additional representation of fuzzy intervals called ACF-representation (using an average cumulative function instead of the membership function), based on possibility theory. We illustrate how to build the new representation and we state its basic properties. The main result is that the Average Cumulative (AC) function can be uniquely defined for any fuzzy interval and it is possible to move from one representation to the others through appropriate transformations. An interesting link can be established between ACF-representation and quantile functions, with a possible statistical interpretation useful in real application. We also recommend a parametric form of the AC function.