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 $\circ$.