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. Let P be a finite ordered set, and let J(P ) be the distributive lattice of order ideals (also called down–sets) of P . For each p ∈ P , define a permutation σp on J(P ) as follows: for every S ∈ J(P ), σp(S) :=  S ∪ {p} if p is minimal in P r S, S r {p} if p is maximal in S, S otherwise. Each of these permutations is an involution. We let Γ(P ) denote the subgroup of the symmetric group Sym(J(P )) generated by all these involutions. Plain curiosity led us to wonder about the structure of these permutation groups. As we shall see, this can be determined quite precisely. As an example, for