Equality of left and right inverses in monoid

This article gives a statement (possibly with proof) of how, if a left-based construction and a right-based construction both exist, they must be equal.
View other such statements

Statement

Suppose $*$ is the associative binary operation of a monoid, and $e$ is its neutral element (or identity element). If an element has both a left and a right inverse with respect to $*$, then the left and right inverse are equal.

Corollaries

• If an element has a left inverse, it can have at most one right inverse; moreover, if the right inverse exists, it must be equal to the left inverse, and is thus a two-sided inverse
• If an element has a right inverse, it can have at most one left inverse; moreover, if the left inverse exists, it must be equal to the right inverse, and is thus a two-sided inverse

Proof

Given: A monoid $S$ with associative binary operation $*$ and neutral element $e$. An element $a$ of $S$ with left inverse $b$ and right inverse $c$.

To prove: $b=c$

Proof: We consider two ways of associating the expression $b*a*c$.

$(b*a)*c=b*(a*c)$ by associativity. The left side simplifies to $e*c=c$ while the right side simplifies to $b*e=b$. Hence, $b=c$.