left inverse

Noun

 * 1)  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.
 * 1)  For a given morphism f : X &rarr; Y, its right inverse (if it has one) is a morphism s : Y &rarr; 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 is preferred if no such $$y$$ exists.

Translations

 * German: Linksinverses