I think a slightly better fix is to change definitions to allow g = { (1, {}) } to be regarded as a left-inverse to g, that is, to allow left-inverses to be partial functions, rather than full functions. The definition still requires they be defined on the image of f, but no choices have to be made on the complement of the image. Probably this breaks some other definitions but it seems intuitively correct to me. It keeps the structure that function B->A could be a left inverse and then only some of them are, rather than limiting them to the functions which are defined only on image(f).

This is kinda nice also because it means that for e.g. the function (a,b) -> (1, 2) given by f(a) = 1, f(b) = 1, you don't need its left inverse to specify that g(2) = a or b, but instead you can have g(2) = {} which doesn't require making any non-canonical choices.

(I'm too sleepy atm to think through this in detail. I might regret this proposal after a nap)

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troethe 5 hours ago | parent [-]

> 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.

	▲	ndriscoll 5 hours ago \| parent \| next [-]
		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".
	▲	ajkjk 5 hours ago \| parent \| prev [-]
		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.