{-1,2}-hypomorphy and hereditarily hypomorphy are the same for posets
Résumé
Let $P$ and $P'$ be two finite posets on the same vertex set $V$. The posets $P$ and $P'$ are {\it 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 $\{-1,2\}$-{\it hypomorphic} if for every subset $X$ of $V$ with $\vert X \vert \in \{2,\vert V\vert -1\}$, the subposets $P(X)$ and $P'(X)$ are isomorphic. P. Ille and J.X. Rampon \cite{Il-Ra} 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.