Inverse element

This is the default notion of inverse element. There is another, more general notion of inverse element in a semigroup, which does not depend on existence of a neutral element

Definition

Definition with symbols

Given a set $S$ with a binary operation $*$ and a neutral element $e$ for $*$, and given elements $a$ and $b$ we say that:

• $b$ is a left inverse to $a$ if $b * a = e$
• $b$ is a right inverse to $a$ if $a * b = e$
• $b$ is an inverse or two-sided inverse to $a$ if $a * b = b * a = e$ (that is, $b$ is both a left and a right inverse to $a$)

An element which possesses a (left/right) inverse is termed (left/right) invertible.

Facts

Equality of left and right inverses

If $*$ is an associative binary operation, and an element has both a left and a right inverse with respect to $*$, then the left and right inverse are equal.

To prove this, let $a$ be an element of $S$ with left inverse $b$ and right inverse $c$. Then, $(b * a) * c = b * (a * c)$ by associativity. The left side simplifies to $e * c = c$ while the right side simplifies to $b * e = b$. Hence, $b = c$.

Some easy 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