Regular Representation wrt Cancellable Element on Finite Semigroup is Bijection
Theorem
Let $\left({S, \circ}\right)$ be a finite semigroup.
Let $a \in S$ be cancellable.
Then:
- the left regular representation $\lambda_a$
and:
- the right regular representation $\rho_a$
of $\left({S, \circ}\right)$ with respect to $a$ are both bijections.
Left Regular Representation wrt Left Cancellable Element on Finite Semigroup is Bijection
Let $\struct {S, \circ}$ be a finite semigroup.
Let $a \in S$ be left cancellable.
Then the left regular representation $\lambda_a$ of $\struct {S, \circ}$ with respect to $a$ is a bijection.
Right Regular Representation wrt Right Cancellable Element on Finite Semigroup is Bijection
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 Cancellable iff Regular Representations Injective, $\lambda_a$ and $\rho_a$ are injections.
We have that $S$ is finite.
From Injection from Finite Set to Itself is Surjection, both $\lambda_a$ and $\rho_a$ are surjections.
Thus $\lambda_a$ and $\rho_a$ are injective and surjective, and therefore bijections.
$\blacksquare$