For a language L ⊆ Σ^*, define the syntactic congruence ≡ of L as the least congruence on Σ^* that saturates L, i.e. :
u ≡ v ⇔ (∀ x, y)[xuy ∈ L ↔ xvy ∈ L].
Now define the Nerode equivalence as the following right congruence :
u ∼ v ⇔ (∀ x)[ux ∈ L ↔ vx ∈ L].
Let [u] be the equivalence class of u with respect to ≡ and 〈u〉 with respect to ∼. Now define i(n) to be the number of different [u] for u of size n, and define j(n) in a similar fashion for ∼.
Now the question is, how do the two functions relate ?
For instance, a standard theorem (Kleene-Schützenberger, I believe) says that i(n) is bounded by a constant whenever j(n) is, and reciprocally.
Question: Is there any other result in this trend? What if one of them is polynomial, for instance?