Given two DFA's accepting the same language, does one have to refine the other?
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?