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

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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.


By Right Cancellable iff Right Regular Representation Injective, $\rho_a$ is an injection.

By hypothesis, $S$ is finite.

From Injection from Finite Set to Itself is Surjection, $\rho_a$ is a surjection.

Thus $\rho_a$ is injective and surjective, and therefore a bijection.