Left Identity in Semigroup may not be Unique

Theorem

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

Let $e_L$ be a left identity of $\struct {S, \circ}$.

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

Proof

Proof by Counterexample

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

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

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

The result follows.

$\blacksquare$

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $4$. Groups: Exercise $4$