Definition:Two-Sided Inverse

Definition

Inverse Mapping

Let $f: S \to T$ and $g: T \to S$ be mappings.

Let:

$g \circ f = I_S$

$f \circ g = I_T$

where:

$g \circ f$ and $f \circ g$ denotes the composition of $f$ with $g$ in either order

$I_S$ and $I_T$ denote the identity mappings on $S$ and $T$ respectively.

That is, $f$ and $g$ are both left inverse mappings and right inverse mappings of each other.

Then:

$g$ is the inverse (mapping) of $f$

$f$ is the inverse (mapping) of $g$.

Inverse Element

The element $y$ is an inverse of $x$ if and only if:

$y \circ x = e_S = x \circ y$

that is, if and only if $y$ is both:

a left inverse of $x$

and:

a right inverse of $x$.

Also see

• Results about inverses can be found here.