Regular Representations on Finite Structure

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.