Condition on Congruence Relations for Cancellable Monoid to be Group

Theorem
Let $\struct {S, \circ}$ be a cancellable monoid whose identity element is $e$.

Then:
 * $\struct {S, \circ}$ is a group


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

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

Hence, let $\RR$ be an arbitrary non-trivial congruence relation.

From Condition for Subgroup of Monoid to be Normal, there exists a normal subgroup $\struct {H, \circ}$ such that:


 * the set of left cosets of $\struct {H, \circ}$ in $\struct {S, \circ}$ form a partition of $S$

and:
 * the set of right cosets of $\struct {H, \circ}$ in $\struct {S, \circ}$ form a partition of $S$

such that the equivalence relations induced by those partitions is $\RR$.

By the definition of normal subgroup, the set of left cosets is the same as the set of right cosets.

It remains to be shown that $\struct {S, \circ}$ is a group.

We already have that

are satisfied by the fact that $\struct {S, \circ}$ is a monoid.

Hence it remains to prove.

Sufficient Condition
Let $\struct {S, \circ}$ be a group.