Monoid

Definition 1. A monoid is a unital semigroup, i.e. a magma $M$ such that:

(Associativity) For all $a$, $b$, $c\in M$,
\[(ab)c = a(bc).\]
(Unitality) There is one $e \in M$ such that for all $a \in M$,
\[ae = ea = a.\]
($e$ is automatically unique by NEU > Proposition 1.)

Proposition 1. Let $M$ be a monoid and $a \in M$. $a$ is left-invertible if and only if $a$ is right-invertible.

Proof. Let $e$ be the neutral element.

If $a$ is left-invertible, there is $x\in M$ such that $xa = e$. We have
\[x(ax) = (xa)x = ex = x,\]
which follows that $ax = e$.
If $a$ is right-invertible, there is $x\in M$ such that $ax = e$. We have
\[(xa)x = x(ax) = xe = x,\]
which follows that $xa = e$.