Right inverse ($f\circ g={\mathrm{id}}_Y$)

A right inverse of a function $f:X\to Y$ is a function $g:Y\to X$ satisfying $f\circ g={\mathrm{id}}_Y$. It forces $f$ to be a surjective function. A surjective linear map onto a finite-dimensional vector space has a linear right inverse: lift a basis of the target and extend by linearity.

The statement that every surjective function has a right inverse is equivalent to the axiom of choice.

Given a surjective function $f:X\to Y$, each fiber of a function $f^{-1}(\{y\})$ is nonempty. The axiom of choice selects $g(y)$ from that fiber for every $y\in Y$, so $f(g(y))=y$ and $g$ is a right inverse. When $Y$ is the empty set, the unique empty function supplies $g$ without a choice.

Conversely, take an indexed family $(X_i{)}_{i\in I}$ of nonempty sets. The projection map from the disjoint union $D=\{(i,x):i\in I,x\in X_i\}$ to $I$ is surjective. A right inverse assigns $i\mapsto (i,x_i)$, selecting one $x_i\in X_i$ for every index. This is a choice function, so the asserted splitting of all surjections gives the axiom of choice.