Additive interactions of n-dimensional random vectors X, as defined by Lancaster, do not necessarily vanish for n≥4 if X consists of two mutually independent subvectors. This defect is corrected and an explicit formula is derived which coincides with Lancaster's definition for n<4. The new definition leads also to a corrected Bahadur expansion and has certain connections to cumulants. The main technical tool is a characterization theorem for the Moebius function on arbitrary finite lattices.