For a monoid $M$ and a subset $S$ of $M$, define the *syntactic congruence* $\equiv_S$ of $S$ as the least congruence on $M$ that saturates $S$, i.e. :
$$u \equiv_S v \Leftrightarrow (\forall x, y)[xuy \in S \leftrightarrow xvy \in S].$$

Now define the *Nerode equivalence* as the following right congruence :
$$u \sim_S v \Leftrightarrow (\forall x)[ux \in S \leftrightarrow vx \in S].$$

Let $[u]_\equiv$ be the equivalence class of $u$ with respect to $\equiv_S$ and $[u]_\sim$ with respect to $\sim_S$.

Now define $i_\equiv (n)$ to be the number of different $[u]_\equiv$ for $u$ of size $n$.

Define $i_\sim(n)$ in a similar fashion.

Now the question is, how do the two $i$ functions relate ?

For instance, a standard theorem says that $i_\sim(n)$ is bounded by a constant whenever $i_\equiv(n)$ is, and reciprocally. Is there any other result in this trend?