Suppose $S$ is a semigroup (or a monoid, or a category), and $X$ is an $S$-set -- that is, a set with an action of $S$. Say that a sub-$S$-set $Y$ is "downward closed" (or maybe "well-generated") if for any $x \in X$ such that $sx \in Y$ for some $s \in S$, we have $x \in Y$.
Has this notion been defined before? I'm really mostly wondering because I'd like to give this a good name.
Notice that if $S$ is a group, this just collapses to any sub-$S$-set since we have inverses. On the other hand, the analogous notion for a module would be the following: a submodule $N$ of $M$ is downward closed if whenever $rm \in N$, $m \in N$. This is related to the notion of a dense submodule, but distinct.