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 $\struct {S / \RR, \circ_\RR}$ 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 $\struct {S, \circ}$.

Let $h_1, h_2 \in H$ and $n_1, n_2 \in N$.

Immediately we have by :
 * $h_1 \circ h_2 \in H$

$h_1 \circ n_1 \in H$.

By :
 * ${h_1}^{-1} \in H$

Hence by :
 * ${h_1}^{-1} \circ \paren {h_1 \circ n_1} = n_1 \in H$

This is a contradiction.

Therefore we must have $h_1 \circ n_1 \in N$.

Similarly, we must have $n_1 \circ h_1 \in N$.

$n_1 \circ n_2 \in H$.

Then by :
 * $\exists h \in H: h \circ \paren {n_1 \circ n_2} = \paren {n_1 \circ n_2} \circ h = e_H$

so $\paren {n_2 \circ h}$ is a right inverse of $n_1$.

If $\circ$ is cancellable, from:
 * $e_H \circ n_2 = n_2 \circ e_H = n_2 \circ \paren {h \circ n_1 \circ n_2}$

we would have:
 * $e_H = n_2 \circ h \circ n_1$

If $\circ$ is commutative:
 * $\paren {n_2 \circ h} \circ n_1 = n_1 \circ \paren {n_2 \circ h} = e_H$

In either case, we see that $\paren {n_2 \circ h} \circ n_1 = e_H$.

So $\paren {n_2 \circ h}$ is also a left inverse of $n_1$.

Thus $n_1$ is invertible, which is a contradiction.

Therefore we must have $n_1 \circ n_2 \in N$.

So far we have:
 * $h_1 \circ h_2 \in H$
 * $h_1 \circ n_1 \in N$
 * $n_1 \circ h_1 \in N$
 * $n_1 \circ n_2 \in N$

Now we show that $\RR$ is a congruence relation on $\circ$.

By definition of $\RR$:
 * $x \mathrel \RR y \iff \set {x, y} \subseteq N \lor \set {x, y} \subseteq H$

and we need to show:
 * $\forall x_1, x_2, y_1, y_2 \in S: \paren {x_1 \mathrel \RR x_2} \land \paren {y_1 \mathrel \RR y_2} \implies \paren {x_1 \circ y_1} \mathrel \RR \paren {x_2 \circ y_2}$

There are $4$ cases:
 * $x_1, x_2, y_1, y_2 \in H$: then $x_1 \circ y_1, x_2 \circ y_2 \in H$, so $\paren {x_1 \circ y_1} \mathrel \RR \paren {x_2 \circ y_2}$

The remaining cases are:
 * $x_1, x_2 \in H, y_1, y_2 \in N$
 * $x_1, x_2 \in N, y_1, y_2 \in H$
 * $x_1, x_2, y_1, y_2 \in N$

In these cases we have:
 * $x_1 \circ y_1, x_2 \circ y_2 \in N$

so $\paren {x_1 \circ y_1} \mathrel \RR \paren {x_2 \circ y_2}$.

Since in each case we have $\paren {x_1 \circ y_1} \mathrel \RR \paren {x_2 \circ y_2}$, $\RR$ is a congruence relation on $\circ$.

Finally we show that the quotient structure $\struct {S / \RR, \circ_\RR}$ is isomorphic to $\struct {\Z_2, \times_2}$, the multiplicative monoid of integers modulo $2$.

We do this by defining a mapping $\phi: S / \RR \to \Z_2$ by:
 * $\map \phi N = 0, \map \phi H = 1$

It is clear that $\phi$ is a bijection.

By writing $H = \eqclass h {\RR}$ and $N = \eqclass n \RR$, where $h \in H$ and $n \in N$, we see that:
 * $H \circ_\RR H = \eqclass h {\RR} \circ_{\RR} \eqclass h {\RR} = \eqclass {h \circ h} {\RR} = H$
 * $H \circ_\RR N = \eqclass h {\RR} \circ_{\RR} \eqclass n {\RR} = \eqclass {h \circ n} {\RR} = N$
 * $N \circ_\RR H = \eqclass n {\RR} \circ_{\RR} \eqclass h {\RR} = \eqclass {n \circ h} {\RR} = N$
 * $N \circ_\RR N = \eqclass n {\RR} \circ_{\RR} \eqclass n {\RR} = \eqclass {n \circ n} {\RR} = N$

and hence:
 * $1 = \map \phi H = \map \phi {H \circ_\RR H} = \map \phi H \times \map \phi H = 1 \times 1$
 * $0 = \map \phi N = \map \phi {H \circ_\RR N} = \map \phi H \times \map \phi N = 1 \times 0$
 * $0 = \map \phi N = \map \phi {N \circ_\RR H} = \map \phi N \times \map \phi H = 0 \times 1$
 * $0 = \map \phi N = \map \phi {N \circ_\RR N} = \map \phi N \times \map \phi N = 0 \times 0$

Thus $\phi$ is an isomorphism.