Nguyen type theorem for extension principle based on a joint possibility distribution

In this paper, first we prove that making abstraction of the output obtained from the interactive extension principle based on a joint possibility distribution, in the case of unimodal fuzzy numbers and when the function that generates the operation is continuous and strictly increasing in each argument restricted to the support of each fuzzy number used in the process, then we can use joint possibility distributions with the property that the left/right side of the output is obtained from the convolution of the values in the left/right side of these fuzzy numbers. Then, considering joint possibility distributions with the aforementioned property, we find an Nguyen type characterization of the level sets of the output based on interactive extension principle, in terms of the level sets of the fuzzy numbers used in the process. These two key results complete well-known results obtained in the case of Zadeh's extension principle and also in the case of triangular norm-based extension principle. As an interesting corollary, in the special case of unimodal fuzzy numbers, the Nguyen theorem can be used to present a new proof concerning necessary and sufficient conditions on the equality of the outputs based on joint possibility distributions, respectively based on Zadeh's extension principle.