Left Identity Element is Idempotent
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Theorem
Let $\struct {S, \circ}$ be an algebraic structure.
Let $e_L \in S$ be a left identity with respect to $\circ$.
Then $e_L$ is idempotent under $\circ$.
Proof
By the definition of a left identity:
- $\forall x \in S: e_L \circ x = x$
Thus in particular:
- $e_L \circ e_L = e_L$
Therefore $e_L$ is idempotent under $\circ$.
$\blacksquare$