To be short, I will abstract a bit from my particular problem.
Let $S$ be a set and $\sim$ be an equivalence relation, defined on that set. Let $R \subseteq (S/\sim) \times (S/\sim)$ be a relation defined on the quotient set $S/\sim$. Let $L \subseteq S$. Can $L$ be closed under $R$? What confuses me here is that $R$ relates an equivalence class to an equivalence class, while $L$ is not an equivalence class.
Thanks in advance for answers.