SO(horn) is the set of boolean queries definable with second-order formulae in normal form such that the quantifier-free part of the formula is in conjunctive normal form with at most one positive instance of a quantified relation per clause. (Atomic queries to the database don't count.)
It was shown in [Grä92] that this class is equal to P.
Those formulae can be made in prenex form where the second order is existential and the first order universal without loss of generality.
No class.