Let $\Gamma$ be a directed graph, and $\Gamma'$ a subgraph. Is there a terminology for the following property?
$\Gamma' \subset \Gamma$ is called ------ if, for all $y \in \Gamma'$ and for all $x$ such that there is an arrow in $\Gamma$ from $x$ to $y$, $x \in \Gamma'$. Moreover, the set of arrows from $x$ to $y$ in $\Gamma'$ is the set of arrows from $x$ to $y$ in $\Gamma$.
Less generally, if $P$ is a poset, is there a term for:
A subposet $P' \subset P$ is called -------- if $P'$ is closed under taking predecessors. (I.e., if $y \in P'$ and $x \leq_P y$, then $x \in P'$.)
A natural term seems to be "predecessor-closed," but I don't want to introduce new terminology if there already exists an accepted term.