Condition on Congruence Relations for Cancellable Monoid to be Group/Counterexample

Theorem
Let $\struct {S, \circ}$ be a monoid which is not cancellable.

Let every non-trivial congruence relation on $\struct {S, \circ}$ be induced by a normal subgroup of $\struct {S, \circ}$.

Then it is not necessarily the case that $\struct {S, \circ}$ is a group.

Proof
Consider the Multiplicative Monoid of Integers Modulo $3$ $\struct {\Z_3, \times_3}$, defined by its Cayley table:

It is noted that $\struct {\Z_3, \times_3}$ is not a cancellable monoid.

Indeed:
 * $0 \times_3 1 = 0$
 * $0 \times_3 2 = 0$

and so $\times_3$ is not cancellable for $0$.

$\struct {\Z_3, \times_3}$ has an identity element $\eqclass 1 3$.

Every non-trivial congruence relation on $\struct {S, \circ}$ is induced by a normal subgroup of $\struct {S, \circ}$.

But $\struct {\Z_3, \times_3}$ is not a group.