Right Identity in Semigroup may not be Unique

Theorem

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

Let $e_R$ be a right identity of $\struct {S, \circ}$.

Then it is not necessarily the case that $e_R$ is unique.

Proof

Proof by Counterexample

Let $\struct {S, \gets}$ be an algebraic structure in which the operation $\gets$ is the left operation.

From Structure under Left Operation is Semigroup, $\struct {S, \gets}$ is a semigroup.

From Element under Left Operation is Right Identity, every element of $\struct {S, \gets}$ is a right identity.

The result follows.

$\blacksquare$