Necessary and sufficient conditions for the equality of interactive and non-interactive extensions of continuous functions

Abstract

In this contribution we find the class of n-dimensional joint possibility distributions with the property that the interactive extension principle coincides with the non-interactive extension principle as long as the interactive operations are determined by continuous functions strictly increasing in each argument. This result completes recent studies by the authors, where the particular case of interactive additions and multiplications versus non-interactive additions and multiplications were investigated. In addition, this time we propose results that also cover the cases when we know the fuzzy numbers only from their membership functions. It means that we eliminated the limitations that appear when we cannot pass from membership function representation to parametric representation of fuzzy numbers. As important new applications, we mention the study on the completely correlated fuzzy numbers. Also of note is that we propose two simple methods to extend bidimensional joint possibility distributions to n-dimensional joint possibility distributions. One method is based on an inductive construction while the other one is based on a pairwise construction.