# 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.

For an integer $$n$$, consider the following DFA $$A_n$$ on the alphabet $$\{a\}$$. The set of states is $$\{q_0,\ldots,q_{n-1}\}$$. The initial state is $$q_0$$. All states are accepting. The transition function is $$\delta(q_i,a) = q_{i+1}$$, where we identify $$q_n$$ with $$q_0$$.
Do the relations of $$A_2$$ and $$A_3$$ refine each other?
• All my automata are over the same alphabet, $\{a\}$. Apr 8 '21 at 17:39