left inverse

English

Noun

left inverse (plural left inverses)

1. (mathematics) A related function that, given the output of the original function, returns the input that produced that output.

   In order for a function to have a left inverse it must be injective.

2. (category theory) For a given morphism f : X → Y, its right inverse (if it has one) is a morphism s : Y → X such that $s\circ f={\mbox{id}}_{X}$.

Usage notes

Given two functions $f:X\to Y$ and $g:Y\to X$, $g$ is the left inverse of $f$ iff for all $x$ in $X$, $g(f(x))=x$. The term is only used if there exists a $y$ in $Y$ such that $f(g(y))!=y$, and an unqualified inverse is preferred if no such $y$ exists.

Antonyms

• right inverse

Hypernyms

• inverse

Translations

Translations

• German: Linksinverses n