Regular Representation wrt Cancellable Element on Finite Semigroup is Bijection

Theorem
Let $$\left({S, \circ}\right)$$ be a finite semigroup.

Let $$a \in S$$ be left cancellable.

Then:


 * The left regular representation $$\lambda_a$$ and
 * and right regular representation $$\rho_a$$

of $$\left({S, \circ}\right)$$ with respect to $$a$$ are both bijections.

Proof
By Cancellable iff Regular Representation Injective, $$\lambda_a$$ and $$\rho_a$$ are injections.

As $$S$$ is finite $$S = \lambda_a \left({S}\right) = \rho_a \left({S}\right)$$.

Thus $$\lambda_a$$ and $$\rho_a$$ are surjections.

Thus $$\lambda_a$$ and $$\rho_a$$ are injective and surjective, and therefore bijections.