# A new language construction

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?

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 $$|\Sigma| \geqslant 2$$.