Some Notes on the Addition of Interactive Fuzzy Numbers.

This paper investigates some fundamental questions involving additions of interactive fuzzy numbers. The notion of interactivity between two fuzzy numbers, say A and B, is described by a joint possibility distribution J. One can define a fuzzy number \(A +_J B\) (or \(A -_J B\)), called J-interactive sum (or difference) of A and B, in terms of the sup-J extension principle of the addition (or difference) operator of the real numbers. In this article we address the following three questions: (1) Given fuzzy numbers B and C, is there a fuzzy number X and a joint possibility distribution J of X and B such that \(X +_J B = C\)? (2) Given fuzzy numbers A, B, and C, is there a joint possibility distribution J of A and B such that \(A +_J B = C\)? (3) Given a joint possibility distribution J of fuzzy numbers A and B, is there a joint possibility distribution N of \((A +_J B)\) and B such that \((A +_J B) -_N B = A\)? It is worth noting that these questions are trivially answered in the case where the fuzzy numbers A, B and C are real numbers, since the fuzzy arithmetic \(+_J\) and \(-_N\) are extension of the classical arithmetic for real numbers.