Counterexample for simple parity automaton reduction

I am dealing with deterministic parity automata and state space reduction (not minimization).

If we define $$\equiv_L$$ to be the equivalence relation that sets two states equal iff starting from those states the automaton recognizes the same $$\omega$$-language, merging those states is in general not an operation that preserves language. The canonical example is a parity automaton with two states $$q_a, q_b$$ that moves to $$q_a$$ whenever reading an $$a$$ and to $$q_b$$ whenever reading a $$b$$. Giving both states a different parity yields a counter example.

Now what if we refine this relation so that $$p \sim q$$ is true iff $$p \equiv_L q$$ and in addition the two states have to have the same color assigned to them. I feel like there has to be a counter example but I can't think of one. That is my question; what is a simple counter example that shows that merging states like this can alter the langauge?

(in case it is not clear what I mean by "merging": merging $$p$$ and $$q$$ means removing $$q$$ from the automaton and redirecting all transitions that would move into $$q$$ to target $$p$$ instead)

• Sorry if I appear pedantic, does your merging $p$ and $q$ also mean redirecting all transitions that would start from $q$ to from $p$? Commented Dec 4, 2018 at 0:51
• No, you want your automaton to stay deterministic. Outgoing transitions from $q$ simply don't exist anymore. Commented Dec 4, 2018 at 8:06

In the original automaton, $$(aab)^\omega$$ has the run $$(103)^\omega$$ which only sees color 1. In the reduced automaton, $$(aab)^\omega$$ has the run $$101(201)^\omega$$, which also visits color 0 infinitely often.