Cancellable Elements of Monoid form Submonoid

Theorem

The cancellable elements of a monoid $\struct {S, \circ}$ form a submonoid of $\struct {S, \circ}$.

Proof

Let $C$ be the set of cancellable elements of $\struct {S, \circ}$.

Obviously $C \subseteq S$.

From Cancellable Elements of Semigroup form Subsemigroup, $\struct {C, \circ}$ is a subsemigroup of $S$.

Let $e_S$ be the identity of $\struct {S, \circ}$.

From Identity of Monoid is Cancellable, $e_S$ is cancellable, therefore $e_S \in C$.

As $e_S$ is the identity of $S$, we have that:

$\forall x \in C: x \circ e_S = x = e_S \circ x$.

Thus $e_S$ is the identity of $\struct {C, \circ}$, which is therefore a monoid.

As $C \subseteq S$, the result follows.

$\blacksquare$