Characterization of the level sets for interactive additions

In this paper, on the set of unimodal fuzzy numbers, first we prove that under some minimal assumptions, it suffices to consider only those joint possibility distributions that satisfy that the left/right side of the fuzzy number generated by the interactive addition based on the given joint possibility distribution, is obtained by evaluating only the left/right side of the summands. Then, we will obtain a representation theorem for the level sets of this interactive addition in terms of the level sets of the summands. These results generalize corresponding results for the special case of triangular-norm-based additions.