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Definition 2024

left_inverse

left inverse

English

Noun

left inverse ‎(plural left inverses)

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

Hypernyms

inverse

Antonyms

right inverse

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.