Three Categories of Set-Valued Generalizations 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 past decades, different authors have independently generalized those definitions to more general contexts, like interval-valued fuzzy sets and Atanassov intuitionistic fuzzy sets. These generalized versions can be introduced either through a list of axioms or in a constructive manner. We can divide them into two further categories: set-valued and point-valued generalized functions. Here, we deal with constructive set-valued generalizations. 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, whereas 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.