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 cancellable.

Then:

the left regular representation $\lambda_a$

and:

the right regular representation $\rho_a$

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

Left Regular Representation wrt Left Cancellable Element on Finite Semigroup is Bijection

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

Let $a \in S$ be left cancellable.

Then the left regular representation $\lambda_a$ of $\left({S, \circ}\right)$ with respect to $a$ is a bijection.

Right Regular Representation wrt Right Cancellable Element on Finite Semigroup is Bijection

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

Let $a \in S$ be right cancellable.

Then the right regular representation $\rho_a$ of $\left({S, \circ}\right)$ with respect to $a$ is a bijection.

Proof

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

We have that $S$ is finite.

From Injection from Finite Set to Itself is Surjection, both $\lambda_a$ and $\rho_a$ are surjections.

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

$\blacksquare$