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 $8.2$