Regular language concatenation with superset Let $A$ be some alphabet.

$A$ itself is a regular language.

$E = A^*$ is regular language over $A$. $E$ is a superset of all languages over $A$, regular or otherwise, i.e $E$ contains every possible string from symbols of $A$.

Now let $L$ be some regular language over $A$.

What is $\tilde{L} = L \cdot E$ ? More specifically, how $\tilde{L}$ and $E$ relate to each other? Are they equal? Is one subset of the other? Something else?

Source of the confusion is the following reasoning: closure under concatenation says that $\tilde{L}$ is a regular language and therefore $\tilde{L} \subseteq E$  (according to the above). On the other hand $\tilde{L}$ seems to contain strings $E$ does not, i.e. those that are formed by concatenating non-empty string from both languages.

Obviously this is wrong. What is going on here?
Simply put, this statement of yours:


  On the other hand $\tilde{L}$ seems to contain strings E does not, i.e. those
  that are formed by concatenating non-empty string from both languages.


is wrong.

As you say, $E$ contains every string containing only symbols from $A$. Since $L$ also only contains strings with symbols from $A$, $\tilde{L}$ must as well. Concatenating strings that have symbols only from $A$ will just get you another string whose symbols only come from $A$.

Your statement is flipped; it is $E$ that contains strings that $\tilde{L}$ may not.