A new language construction

Question

I'm interested if the following language construction was studied. Let $\Sigma$ be an alphabet and $\Sigma^*$ is the set of all words over $\Sigma$. Consider a directed graph $G=\langle\Sigma^*,E\rangle$ together with a subset $S\subseteq\Sigma^*$ of its vertices. Now, define the language $L=\langle\Sigma,E,S\rangle$ consisting of all words reachable from $S$ in $G$. That is if $w\in L$ then $w\in S$ or there is a word $u\in S$ and some path from $u$ to $w$ in $G$. Can we describe $L$ using the Chomsky hierarchy?

Answer 1

No, you can obtain any submonoid $M$ of $\Sigma^*$ using your construction. Just take $S = \{1\}$ and $E = \{(x, xy) \mid x, y \in M\}$. Then $L = M$.

There are submonoids of $\Sigma^*$ at each level of the Chomsky hierarchy if $|\Sigma| \geqslant 2$.