Necessary and sufficient conditions for the equality of the interactive and non-interactive sums of two fuzzy numbers

This study aims to answer the open question proposed by Carlsson et al. (2004) [4], which asks whether it is possible to find the properties needed by a joint possibility distribution function such that the interactive sum with respect to this function and the standard sum of two fuzzy numbers will coincide. We provide the necessary and sufficient conditions such that this equality would hold. In addition, we find the necessary and sufficient conditions such that the interactive and standard sums will coincide at some point in the domain. In order to obtain an interactive addition that does not coincide with the standard addition, we show that we need to search for a joint possibility distribution that coincides as little as possible with the independent joint possibility distribution throughout the so-called diagonal of the two fuzzy numbers. As simple consequences, we deduce similar results for triangular norm-based additions of fuzzy numbers.