A permutation group determined by an ordered set

Let P be a finite ordered set, and let J(P) be the distributive lattice of order ideals of P. The covering relations of J(P) are naturally associated with elements of P; in this way, each element of P defines an involution on the set J(P). Let Γ(P) be the permutation group generated by these involutions. We show that if P is connected then Γ(P) is either the alternating or the symmetric group. We also address the computational complexity of determining which case occurs.