I have a logical question that I can’t quite crack:

Given two automata accepting the same language $L$, does one have to refine the other?

In other words, if $A_1$ and $A_2$ both accept $L$, with associated equivalence relations $R_{A_1}$ and $R_{A_2}$, does $R_{A_1}$ have to refine $R_{A_2}$, or vice versa?

I am leaning toward the answer yes because if we have a regular language $L$ which is accepted by an automaton $A$, we can show that the relation $R_A$ refines the relation $R_L$, meaning $R_A sqsubseteq R_L$,

which means that both $R_{A_1} sqsubseteq R_L$ and $R_{A_2} sqsubseteq R_L$.

We are currently studying the Myhill-Nerode Theorem, so I’m guessing it has something to do with it.

I’ve tried combining few theorems together, but came out empty.