Recall that a monoid $M$ is a set equipped with an associative binary operation $* : M \times M \to M$ and an identity element $e \in M$ such that for any $m \in M$, $e*m = m = m*e$. We sometimes omit $*$ and concatenate the symbols.

Inverses

A left (resp. right) inverse of $m \in M$ is an element $n \in M$ such that $nm = e$ (resp. $mn = e$).

Proposition. (Equality of left and right inverses) If both left and right inverses exist for an element, then they are equal. That is if $am = e$ and $mb = e$ then $a = b$.

Proof. Let $m \in M$ and $a,b \in M$ such that $am = e = mb$. Then we have $a = ae = a(mb) = (am)b = eb = b$.

Corollary. In the context of monoids:

• If left and right inverses of an element exist, then it is a two sided inverse and is unique.
• If an element has two distinct left (resp. right) inverses, it cannot have a right (resp. left) inverse.