Definition:Left Inverse Mapping

Definition
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 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.