Cancellable Elements of Semigroup form Subsemigroup

Theorem
Let $\struct {S, \circ}$ be a semigroup.

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

Then $\struct {C, \circ}$ is a subsemigroup of $\struct {S, \circ}$.

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

Let $x, y \in C$.

Then $x$ and $y$ are both left cancellable and right cancellable.

Thus by Left Cancellable Elements of Semigroup form Subsemigroup:


 * $x \circ y$ is left cancellable

and by Right Cancellable Elements of Semigroup form Subsemigroup:


 * $x \circ y$ is right cancellable.

Thus $x \circ y$ is both left cancellable and right cancellable, and therefore cancellable.

Thus $x \circ y \in C$.

Thus $\struct {C, \circ}$ is closed and is therefore by the Subsemigroup Closure Test a subsemigroup of $\struct {S, \circ}$.