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

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Theorem

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.


Proof

By Left Cancellable iff Left Regular Representation Injective, $\lambda_a$ is an injection.

By hypothesis, $S$ is finite.

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

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

$\blacksquare$