So glad to find this place. I have been struggling for quite a while with this given question and i am not sure how to fully address it.
The question: $L_1$ and $L_2$ are regular languages over the same $\Sigma$. prove that the following language is regular as well:
$L^\frown \:=\left\{\:w\:\in L_1 | \:w=\sigma _1\mu _1\xi _1\sigma \:_2\mu \:_2\xi \:_2...\sigma \:_n\mu \:_n\xi \:_n\right\}$ Where $n \geq 0$, for every i:$\left(1\:\le i\:\le n:\:\sigma _i\mu _i\xi _i\:\in \Sigma \right)$, $\mu_1\mu_2...\mu_n \in L_2$.
Prove using: 1)Closure properties. 2)Building an multipication automaton
My try:
1)I do not understand how to define the homomorphism from which I can deduct using closure properties that the given language is indeed regular.
2)Since both $L_1$ and $L_2$ are regular, There exist two deterministic finite automatas that can accept them: $A^i\:=\:\left(\Sigma ,\:Q^i,q_0^i,F^i,\delta ^i\right)$ ($i\:\in \left\{\:1,2\right\})$. So, let L be the automaton that accepts those two languages: let L be $L\:=\:\left(\Sigma \:,\:Q^1\:X\:Q^2,\left(q_0^1,q_0^2\right),F^1X\:F^2,\delta \:\right)$. However, here I don't know what its transition function should be
Would really appreciate your help with it, if possible with explanations to understand so I'll be able to overcome similar questions in the future.
Thank you so much!