> your fix kinda breaks a lot of the structure of algebra in other ways

What are you referring to here in particular?

I think the property of `g` to be a well defined function is a lot more important than for its domain to be `B`, when `f(A)` is enough to make the composition well defined.

Typically one defines relations before functions anyway (unless you're doing type theory/programming, in which case types matter), and relations also offer a fix (the empty relation from the empty set is left inverted by the empty relation from the target to the empty set).

Most algebra books I've read are either explicitly or at least implicitly setting up structural analogies to introduce categories, where your A and B are indeed fixed/"typed".

sorry I edited my comment because I wasn't quite sure what I meant by that either. I am too sleepy.

But, I disagree. partial functions B->A are just a superset of functions B->A, so requiring g to be partial is just relaxing a requirement. Whereas requiring g: im(A)->A adds an additional restriction, potentially breaking any place that concept of a left inverse is used. Invariably someone will just define a new thing equivalent to what is normally called the left inverse because they still need the concept for some reason.