$A$ nonregular language, is $A\cup B$ nonregular?

Question

I was reading this question; regarding whether $L=A\cup B$ is nonregular if $A$ is. As is rightly pointed out in that answer, there is no simple rule. But what if we imposed the following conditions:

1. $A \cap B=\emptyset$
2. $A\neq B^C$ and $B \neq A^C$

The examples I know of where $L$ fails to be nonregular is where $B$ somehow "makes up" for the nonregularity of $A$, such as setting $B$ as the language of all strings, or the complement of $A$.

But with these new conditions, is there any way for $B$ to turn the union $A\cup B$ into a regular language?

EDIT: Actually, if $L=\{0^n1^m|n,m\geq 0\}$, then

$$L=\{0^n1^n|n\geq 0\} \cup \{0^n1^m|n\neq m\}$$

and both the conditions above are fulfilled. So $L$ can still be regular. Sorry :)

Answer 1

Here is one result in the direction you're looking into:

Suppose that $A,B,C$ are languages such that $A$ is a non-regular subset of the regular language $C$, and $B$ is disjoint from $C$. Then $A \cup B$ is non-regular.

For the proof, consider the intersection of $A \cup B$ and $C$. Details left to the reader.

The question you link to concerns the language $$ \{ a^p : p \text{ is prime} \} \cup \{ w : |w| \text{ is even} \} = \{ a^p : p \text{ is an odd prime} \} \cup \{ w : |w| \text{ is even} \}. $$ We can write this in the form $A \cup B$, where $A = \{ a^p : p \text{ is an odd prime}\}$ and $B = \{ w : |w| \text{ is even}\}$. Let $C = \{ w : |w| \text{ is odd} \}$. Applying the result above, we deduce that $A \cup B$ is non-regular (assuming we already know that $A$ is non-regular).