Two-sided inverse

Definition: Two-sided inverse

If is a function, then is a two-sided inverse of if and .

If [math]f [/math] has a two-sided inverse, it must be unique, so we are justified in writing the two-sided inverse of [math]f [/math]. We also write to denote the inverse of [math]f [/math] if it exists.

If and then [math]f [/math] is a left inverse of [math]g [/math], because

For the same reason, [math]g [/math] is a right inverse of [math]f [/math].

However, [math]f [/math] is not a right inverse of [math]g [/math] (nor is [math]g [/math] a left inverse of [math]f [/math]) because

Finally, if [math]h := [/math] and [math]i := [/math] , then [math]h [/math] and [math]i [/math] are two-sided inverses of each other, because

and