Example of a Monoid with two distinct congruence relation

Question

I am looking for an example of a monoid $M$ with identity element $e$ and two distinct congruence relations $R \neq R'$ on $M$.

I feel that I need a non-cancellative Monoid, but I am struggling here.

Answer 1

Well, for each monoid $M$, the identity relation $\{(a,a)\mid a\in M\}$ and the all-relation $\{(a,b)\mid a,b\in M\}$ are congruence relations, the trivial ones.

Answer 2

Although Wuestenfux's answer is perfectly correct, if you want to avoid trivial congruences, start with any nontrivial monoid $M$ and consider the monoid $M \times M$. Now define \begin{align} (x_1, x_2)\sim_1 (y_1, y_2) &\iff x_1 = y_1 \\ (x_1,x_2) \sim_2 (y_1, y_2) &\iff x_2 = y_2 \end{align} Then both $\sim_1$ and $\sim_2$ are congruences on $M \times M$, and you can choose $M$ (and hence $M \times M$) to be cancellative if you wish.