Definition:Left Inverse Mapping

Definition
Let $S, T$ be sets where $S \ne \O$, i.e. $S$ is not empty.

Let $f: S \to T$ be a mapping.

Let $g: T \to S$ be a mapping such that:
 * $g \circ f = I_S$

where:
 * $g \circ f$ denotes the composite mapping $f$ followed by $g$;
 * $I_S$ is the identity mapping on $S$.

Then $g: T \to S$ is called a left inverse (mapping).

Also see

 * Injection iff Left Inverse, which demonstrates that $g$ can not be defined unless $f$ is an injection.


 * Definition:Right Inverse Mapping

In the context of abstract algebra: from which it can be seen that a left inverse mapping can be considered as a left inverse element of an algebraic structure whose operation is composition of mappings.
 * Definition:Right Inverse Element
 * Definition:Left Inverse Element