On product hierarchies of automata

Abstract

We have seen that, with respect to homomorphic realization, the ν _i —products behave in a way similar to the α _i —products on classes satisfying the Letičevskiî criterion or not satisfying the Letičevskiî criteria. In particular, a class K is homomorphically ν ₃—complete if and only if it satisfies the Letičevskiî criterion. As opposed to the α _i —products, the ν _i —hierarchy is proper on classes with the semi-Letičevskiî criterion. This holds also for homomorphic realization.

To give a summary of the rest of our comparison results, take any two of our product notions, the “β—product” and the “γ—product”, say. Define

γ if

holds for all K. Similarly let

γ if we have

for all K. We obtain two poset structures whose exact diagrams are given in the figures below. The bottom is the quasi-direct product in both cases, for it is obvious that

and

ν ₁, henceforth also

and

. (We write β<γ if β≤γ but γ≰β.)

Recently it has been shown by the first two authors that there is a class K satisfying the Letičevskiî criterion but which is not homomorphically ν ₂-complete.

Research supported by Alexander von Humboldt Foundation