I'm reading through Dummit and Foote, and they have the following statements about functions and inverses.
Given a map $f: A\to B$ :
- The map $f$ is injective if and only if $f$ has a left inverse.
- The map $f$ is surjective if and only if $f$ has a right inverse.
- The map $f$ is a bijection if and only if there exists $g: B\to A$ such that $f\circ g$ is the identity map on $B$ and $g\circ f$ is the identity map on $A$.
Where $f$ has a left inverse if there exists a function $g: B\to A$ such that $g\circ f$ is the identity on $A$ and $f$ has a right inverse if there exists a function $h: B\to A$ such that $f\circ h$ is the identity on $B$.
The relation between inverses and injectiveness/surjectiveness/bijectiveness make sense to me intuitively. But in particular, the bijection case requires that the right and left inverse be the same. Does this mean that there exists a function $f$ that is not a bijection which has distinct left and right inverses? Or would such a function require a quantifier of the domain of its inverses? The only thing I can tell is that where the domain of the left and right inverses overlap, they must be equal. If the overlap of the domains is $B'$, then $f$ must be injective on $B'$ (because it has a left inverse in that domain), and $f$ must also be surjective (because it has a right inverse in that domain). Therefore, $f$ is bijective on $B'$ and $\forall a\in B', g(a)=h(a)$. What else can be concluded?