Left Identity Element is Idempotent

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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 $\circ$.


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 $\circ$.


Also see