Right Cancellable iff Right Regular Representation Injective

Theorem
Let $\left({S, \circ}\right)$ be a semigroup.

Then $a \in S$ is right cancellable iff the right regular representation $\rho_a \left({x}\right)$ is injective.

Proof
Suppose $a \in S$ is right cancellable.

Then $x \circ a = y \circ a \implies x = y$.

From the definition of the right regular representation, $\rho_a \left({x}\right) = x \circ a$.

Thus:
 * $\rho_a \left({x}\right) = \rho_a \left({y}\right) \implies x = y$

and so the right regular representation is injective.

Suppose $\rho_a \left({x}\right)$ is injective.

Then:
 * $\rho_a \left({x}\right) = \rho_a \left({y}\right) \implies x = y$

From the definition of the right regular representation:
 * $\rho_a \left({x}\right) = a \circ x$

Thus:
 * $a \circ x = a \circ y \implies x = y$

and so $a$ is right cancellable.