I know that if $L_{1},L_{2}$ are regular languages then so is $L_{1}\cap L_{2},L_{1}\cup L_{2}$ are regular languages, I also know that $L$ is regular $\implies L^{c}$ is regular .
It is easy to generalize in this manner:
Let $L_{1},...L_{n}$ be regular languages and $x\in\{0,1\}^{n}$ a vector, then $L_{1}^{x_{1}}\cap...\cap L_{n}^{x_{n}}, L_{1}^{x_{1}}\cup...\cup L_{n}^{x_{n}}$ is regular, where $L^{0}$ denotes $L^{c}$ and $L^{1}$ denotes $L$, I can also generalize by changing some of the $\cap$ to $\cup$ .
My question is: How can this be generalized ?
It seems that both statements have some kind of Boolean operations in the scene that given $L_{1},...,L_{n}$ I can represent those languages with binary vectors (I also noted that $A\cup B=(A^{c}\cap B^{c})^{c}$ so the two generalizations I gave seem to be the equivalent to each other)