A hierarchy of shift equivalent sofic shifts

We define new subclasses of the class of irreducible sofic shifts. These classes form an infinite hierarchy where the lowest class is the class of almost finite type shifts introduced by B. Marcus. We give effective characterizations of these classes with the syntactic semigroups of the shifts. We prove that these classes define invariants shift equivalence (and thus for conjugacy). Finally, we extend the result to the case of reducible sofic shifts.