{-1,2}-hypomorphy and hereditary hypomorphy coincide for posets

Let P and P' be two finite posets on the same vertex set V. The posets P and P' are hereditarily hypomorphic if for every subset X of V, the induced subposets P(X) and P'(X) are isomorphic. The posets P and P' are {-1,2}-hypomorphic if for every subset X of V, |X| in {2,|V|-1}, the subposets P(X) and P'(X) are isomorphic. P. Ille and J.X. Rampon showed that if two posets P and P', with at least 4 vertices, are {-1,2}-hypomorphic, then P and P' are isomorphic. Under the same hypothesis, we prove that P and P' are hereditarily hypomorphic. Moreover, we characterize the pairs of hereditarily hypomorphic posets.