# automata – Given two DFA’s accepting the same language, does one have to refine the other?

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.