Which closure properties are always valid between regular, context-free and non context-free languages?

Question

I am making a scheme that respresents some closure properties (union, intersection, complement and concatenation) for regular languages, context-free languages, decidable languages and RE languages. Below you can see my scheme.

I am trying to find all possible internal closure properties of these different languages that will always be valid. E.g., I know that $$\text{Regular language } \cup \text{ CFL } = \text{ CFL } $$

$$\text{Regular language } \cap \text{ CFL } = \text{ CFL } $$ $$\text{Regular language } ^* \text{ CFL } = \text{ CFL } $$

What other internal properties are there between these languages?

Answer 1

I am afraid that there is no better general result than the obvious one.

Assume that $\mathcal K$ and $\mathcal L$ are two families of languages with $\mathcal K\subset \mathcal L$ such that $\mathcal L$ is closed under the operation $\circ$. Then for $K\in\mathcal K$ and $L\in \mathcal L$ we have $K\circ L\in \mathcal L$.

This is obvious, as both $K,L\in \mathcal L$. Usually we can choose simple examples that we can't do better. For instance, choose a "difficult" language in $\mathcal L$, so $L\in \mathcal L \setminus \mathcal K$. Then $L=\varnothing \cup L$, $L=\Sigma^*\cap L$, and $L=\{\lambda\}\cdot L$ show that we cannot do better than $\mathcal L$.