# Identity is only Idempotent Cancellable Element

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## Theorem

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.

## Proof

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 $x = e_S$.

$\blacksquare$

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 8$: Compositions Induced on Subsets: Theorem $8.2$