When is a Category of Many-Sorted Partial Algebras Cartesian-Closed?

In this paper we study the cartesian closedness of the five most natural categories with objects all partial many-sorted algebras of a given signature. In particular, we prove that, from these categories, only the usual one and the one having as morphisms the closed homomorphisms can be cartesian closed. In the first case, it is cartesian closed exactly when the signature contains no operation symbol, in which case such a category is a slice category of sets. In the second case, it is cartesian closed if and only if all operations are unary. In this case, we identify it as a functor category and we show some relevant constructions in it, such as its subobjects classifier or the exponentials.