Additions of completely correlated fuzzy numbers

In this paper we consider additions of interactive fuzzy numbers. The interactivity relation between fuzzy numbers are defined by their joint possibility distribution. We show that Nguyen´s theorem remains valid in this environment. We give explicit formulas for the γ-level sets of the extended sum of two completely correlated fuzzy numbers. We show that the interactive and the non-interactive sums have the same membership function for any pair of completely positively correlated fuzzy numbers. Finally, we prove that (i) the interactive sum of two completely negatively correlated fuzzy numbers A and B with A(x) = B(-x) for all x ∈ R, is (crisp) zero; (ii) the interactive difference of two completely positively correlated fuzzy numbers A and B with identical membership function, that is, a(x) - b(x) for all x ∈ R, is (crisp) zero.