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.