Condition for Partition between Invertible and Non-Invertible Elements to induce Congruence Relation on Monoid

Theorem
Let $\struct {S, \circ}$ be a monoid whose identity element is $e$ such that $\struct {S, \circ}$ is specifically not a group.

Let $\struct {H, \circ}$ be the subgroup of $\struct {S, \circ}$ consisting of its invertible elements.

Let $N$ be the set of non-invertible elements of $\struct {S, \circ}$.

Let $\RR$ be the equivalence relation induced by the partition $\set {N \mid H}$.

Let either:
 * $\circ$ be a cancellable operation

or:
 * $\circ$ be a commutative operation.

Then:
 * $\RR$ is a congruence relation on $\circ$

and:
 * the quotient structure $\struct {E / \RR, \circ_\RR}$ is isomorphic to $\struct {\Z_2, \times_2}$, the multiplicative monoid of integers modulo $2$

Proof
First we confirm from Invertible Elements of Monoid form Subgroup that $\struct {H, \circ}$ is in fact a subgroup of $\struct {S, \circ}$.