On Gradual Sets, Hesitant Fuzzy Sets, and Representation Theorems

We analyze the connection between two perspectives when defining fuzzy sets: the viewpoint of mappings and the viewpoint of families of level cuts. This analysis is mathematically supported by the framework of a categorical adjunction, which serves as a dictionary between these two perspectives. We prove that hesitant fuzzy sets and gradual sets are strongly related through this connection. This allows concepts and operations to be transferred from one class to the other, and vice versa. Concretely, as an application, we provide lattice operations on gradual sets, compatible with Zadeh's max-min operations on fuzzy sets, when considering them as families of level cuts. We discuss the well-known representation theorem for fuzzy sets within this framework, and we show that the representation of fuzzy sets as gradual sets depends on the chosen embedding of fuzzy sets as hesitant fuzzy sets. Hence, distinct embeddings yield diverse representations of fuzzy sets as collections of subsets. Furthermore, we extend this methodology to include other classes of extended fuzzy sets. As a consequence, a representation theorem for interval-valued fuzzy sets is provided.