We study the relationship between standard alphabets over the equality atoms, and those which can be made homogeneous using finitely many relations and functions. We leave open the problem whether these notions coincide.
Fix attention on equality atoms.
Given an orbit-finite, deterministic automaton on the alphabet of atoms, a Brzozowski phase amounts to:
- reversing the direction of all transition arrows and swapping initial and accepting states,
- determinising the result, and
- cutting to the reachable part.
We study a derivation quasi-order between between alphabets. Whenever is derived from and is non-standard then is non-standard too. We identify also a class of minimal non-standard alphabets wrt. the quasi-order.