Three categories of set-valued generalisations from fuzzy sets to interval-valued and Atanassov intuitionistic fuzzy sets

Many different notions included in the fuzzy set literature can be expressed in terms of functionals defined over collections of tuples of fuzzy sets. During the last decades, different authors have independently generalised those definitions to more general contexts, like interval-valued fuzzy sets and Atanassov intuitionistic fuzzy sets. These generalised versions can be introduced either through a list of axioms or in a constructive manner. We can divide them into two further categories: setvalued and point-valued generalized functions. Here we deal with constructive set-valued generalisations. We review a long list of functions, sometimes defined in quite different contexts and we show that we can group all of them into three main different categories, each of them satisfying a specific formulation. We respectively call them the set-valued extension, the max-min extension and the max-min-varied extension. We conclude that the set-valued extension admits a disjunctive interpretation, while the max-min extension can be interpreted under an ontic perspective. Finally, the max-min varied extension provides a kind of compromise between both approaches.