Definition 1: Consider $S = \{ a, a^2, \dots, a^n, \cdots \}$ is a semigroup under the operation $$ a^i a^j = a^{i+j} \; \; \forall i,j \in \mathbb N$$ is called a free monogenic semigroup.
Definition 2: A equivalence relation $\rho$ on a set $S$ is a congruence if $$(\forall s, t, a\in S) (s, t)\in \rho \implies (sa, ta) , (as, at)\in \rho.$$
Definition 3: A congruence relation on $S$ is said to be group congruence if $S/ \rho$ is group under the binary operation $$(a\rho) (b\rho) = ab\rho$$
- Define a relation $\rho_n$ on $S$ by $$ a^r \rho_n a^s \Longleftrightarrow \; r \equiv s (\text{mod n)} $$
I have prove that $\rho_n$ is a group congruence and $S/ \rho_n$ isomorphic to $\mathbb Z_n$
Question
If $\rho$ is a group congruence on $S$, then $\rho = \rho_n$ for some $n \in \mathbb N$.
My attempt is
Suppose $\rho$ is a group congruence on $S$, then define $$n = \text{min} \{ | r-s| : (a^r, a^s) \in \rho \backslash 1_S \} $$ where $1_S$ is the diagonal realtion on $S$. Clearly if $|r-s| < n$, then $(a^r, a^s) \notin \rho$, so $a\rho$, $a^2\rho$, $\cdots, a^{n}\rho$ are all distinct classes.I want to show that $(a, a^n) \in\rho, \; (a^2, a^{n+1} ) \in \rho$ and so on. Also $\exists r \in \mathbb N$ such that $(a^r, a^{n+r}) \in \rho$. I am stuck here.
Please help.