# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Corollary 2.1.2.4. Let $M$ be a nonunital monoid. Then an element $e \in M$ is a unit if and only if the following conditions are satisfied:

$(i)$

The element $e$ is idempotent: that is, we have $ee = e$.

$(ii)$

The element $e$ is left cancellative: that is, the function $x \mapsto ex$ is a monomorphism from $M$ to itself.

$(iii)$

The element $e$ is right cancellative: that is, the function $x \mapsto xe$ is a monomorphism from $M$ to itself.