Identity is only Idempotent Cancellable Element

Theorem

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

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

$\blacksquare$

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