Identities are Idempotent
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Theorem
Right Identity Element is Idempotent
Let $\struct {S, \circ}$ be an algebraic structure.
Let $e_R \in S$ be a right identity with respect to $\circ$.
Then $e_R$ is idempotent under $\circ$.
Left Identity Element is Idempotent
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 $\circ$.
Identity Element is Idempotent
Let $\struct {S, \circ}$ be an algebraic structure.
Let $e \in S$ be an identity with respect to $\circ$.
Then $e$ is idempotent under $\circ$.