f-inclusion indexes between fuzzy sets

We introduce the notion of f-inclusion, which is used to describe different kinds of subsethood relations between fuzzy sets by means of monotonic functions f : [0, 1] → [0, 1]. We show that these monotonic functions can be considered indexes of inclusion, since the greater the function considered, the more restrictive is the relationship. Finally, we propose a general index of inclusion by proving the existence of a representative f-inclusion for any two ordered pairs of fuzzy sets. In such a way, our approach is different from others in the literature in no taking a priori assumptions like residuated implications or t-norms.