DPA · ω-regular
Eventually always b, by priority
The same FG b, deterministic, with α as a per-state priority instead of a set. Parity is where determinism stops costing expressive power.
The author’s examples, run
(b) → accepta(b) → acceptab(b) → acceptaaaa(b) → accept(a) → reject(ab) → rejectbbb(ab) → reject