Additive similarity and dissimilarity measures

We develop a detailed formal study of the different variants of the notions of similarity and dissimilarity between fuzzy sets under the assumption of additivity . A measure of (dis)similarity is said to be additive when it can be decomposed as the sum of the particular (dis)similarities of the compared memberships over the different elements of the universe. Some formal relations between existing formalizations of the notions of similarity and dissimilarity arise under this additional assumption. We conclude that, in this particular setting, one may basically distinguish between two main categories of comparison measures in the literature: those that depend on both the intersection and the difference of the compared items and those that only depend on their difference.