Definition:Right 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:

$f \circ g = I_T$

where:

• $f \circ g$ denotes the composite mapping $g$ followed by $f$;
• $I_T$ is the identity mapping on $T$.

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

Also see

• Surjection iff Right Inverse, which demonstrates that $g$ can not be defined unless $f$ is a surjection.

• Left Inverse Mapping

In the context of abstract algebra:

• Left inverse element
• Right inverse element

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.

Sources

• George McCarty: Topology: An Introduction with Application to Topological Groups (1967): $\text{I}$
• Ian D. Macdonald: The Theory of Groups (1968): Appendix
• H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability (1996): Appendix $\text{A}.7$