Let $A$ be an irreducible nonnegative $N\times N$ integer matrix with constant row sum $D$. Let $A_1, \dots, A_D$ be nonnegative integer matrices, each with constant row sum $1$, such that $\sum_k A_k = A$. Each $A_k$ defines a mapping $f_k: [N] \to [N]$ where $f_k(i) = j$ iff $(A_k)_{ij} = 1$. Let $S \subset [N]^{[N]}$ denote the transformation semigroup generated by the $f_k$'s; equivalently, the finite semigroup of matrices with row sum $1$ generated by the $A_k$'s. Let $S^*$ denote the minimal ideal of $S$. As matrices, the elements $S^*$ are the matrices with minimal rank $r$; as mappings, these are the $f$ with $|f([N])|=r$.
I am interested in bounds on the possible values of $r$ and $|S^*|$ in terms of $D$ and $N$. There are some trivial bounds, but I am wondering if someone who knows e.g. Krohn-Rhodes theory could point me in a suitable direction.
I am particularly interested in using such bounds to address the following question. For $f \in S^*$, let $\alpha_f$ be the partition of $[N]$ where $\alpha_f(i) = \alpha_f(j) \Leftrightarrow f(i) = f(j)$. Let $\alpha^* = \bigvee_{f \in S^*} \alpha_f$, the coarsest common refinement of the $\alpha_f$'s (I'm aware that in some settings, the opposite convention for $\vee$ vs. $\wedge$ is used). So $\alpha^*(i) = \alpha^*(j)$ iff $\alpha_f(i) = \alpha_f(j)$ for all $f \in S^*$.
I would really like to show that there is a choice of the $A_k$'s such that $\alpha^*$ is not the discrete partition, i.e. such that $|\alpha^*| \leq N-1$. (This is equivalent to Trahtman's Road Colouring Theorem, but I would like to avoid using that result, as I'm trying to generalize the result to a setting where his method does not work well.)
