A Lattice Structure on Hesitant Fuzzy Sets

In this article, we deal with the lattice-compatibility between several classes of extended fuzzy sets. Concretely, we treat the problem of finding a lattice structure on set-valued fuzzy sets ( $\operatorname{SVFS}$ s) whose restriction to interval-valued fuzzy sets ( $\operatorname{IVFS}$ s) and (type-1) fuzzy sets ( $\operatorname{FS}$ s) match Zadeh's classical lattice operations. A prominent approach to this problem was given by Torra by means of the so-called hesitant fuzzy sets ( $\operatorname{HFS}$ s). Nevertheless, despite their usefulness in group decision-making problems, it is well-known that Torra's operations do not produce a lattice. Here, we mend partially this handicap by giving two lattice orders. Each of them preserves one of the Torra's operations and, additionally, reduces to Zadeh's orders on $\operatorname{FS}$ s and on $\operatorname{IVFS}$ s. As a counterpart, they cannot be defined on the whole class of $\operatorname{HFS}$ s, or $\operatorname{SVFS}$ s. Finally, we provide a full answer combining both orders. We define a partial order, that we call the symmetric order, on the whole class of nonempty subsets of [0,1]. This order extends the usual ones on [0,1] and on closed intervals of [0,1]. As a consequence, we find a lattice structure on $\operatorname{HFS}$ s whose restriction to $\operatorname{FS}$ s and $\operatorname{IVFS}$ s reduces to Zadeh's operations.