$\downarrow$-posets

We investigate a certain class of posets arising from semilattice actions. Let $S$ be a semilattice with identity. Let $S$ act on a set $C$. For $c,d\in C$ put $c\leq d$ iff there is some $s\in S$ with $ds=c$. Then $(C,\leq)$ is a poset. Let's call the posets that arise in this way $\downarrow$-posets. We give a reasonable second order characterization of $\downarrow$-posets and show that there is no first order characterization.