Identity of Monoid is Cancellable

Theorem

Let $\struct {S, \circ}$ be a monoid.

Let $e$ be an identity element of $\struct {S, \circ}$.

Then $e$ is a cancellable element of $\struct {S, \circ}$.

Proof

Let $x, y \in S$ such that $x \circ e = y \circ e$.

Then, by definition of identity element:

$x = x \circ e = y \circ e = y$

Thus $x = y$.

$\blacksquare$

Also see

• Invertible Element of Monoid is Cancellable