Identity is only Idempotent Cancellable Element

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Let $e_S$ is the identity of an algebraic structure $\struct {S, \circ}$.

Then $e_S$ is the only cancellable element of $\struct {S, \circ}$ that is idempotent.


By Identity Element is Idempotent, $e_S$ is idempotent.

Let $x$ be a cancellable idempotent element of $\struct {S, \circ}$.

\(\ds x \circ x\) \(=\) \(\ds x\) $x$ is idempotent
\(\ds \) \(=\) \(\ds x \circ e_S\) Definition of Identity Element

So $x \circ x = x \circ e_S$.

But because $x$ is also by hypothesis cancellable, it follows that $x = e_S$.