Left Identity in Semigroup may not be Unique
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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
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 $4$