Definition:Left Inverse Mapping

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

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

Let $g: T \to S$ be a mapping such that:

$g \circ f = I_S$


$g \circ f$ 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

In the context of abstract algebra:

from which it can be seen that a right inverse mapping can be considered as a right inverse element of an algebraic structure whose operation is composition of mappings.