Left Identity in Semigroup may not be Unique

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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 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.