Regular Representations wrt Element are Permutations then Element is Invertible

Theorem
Let $\struct {S, \circ}$ be a semigroup.

Let $\lambda_a: S \to S$ and $\rho_a: S \to S$ be the left regular representation and right regular representation with respect to $a$ respectively:

Let both $\lambda_a$ and $\rho_a$ be permutations on $S$.

Then there exists an identity element for $\circ$ and $a$ is invertible.