Right Regular Representation wrt Right Cancellable Element on Finite Semigroup is Bijection
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Theorem
Let $\struct {S, \circ}$ be a finite semigroup.
Let $a \in S$ be right cancellable.
Then the right regular representation $\rho_a$ of $\struct {S, \circ}$ with respect to $a$ is a bijection.
Proof
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.
$\blacksquare$