As usual in field theory, the convention to consider p / q as an abbreviation for p · q−1 was used in subsequent work on meadows (see e.g. [2,5]). This convention is no longer satisfactory if partial variants of meadows are con- sidered too, as is demonstrated in [3].
So, as I’ve stated many times, I talked about convention and indicated you can use whatever terms you want. In the paper quoted above they acknowledge what the convention is. That is that division is multiplication by the inverse. They are arguing that it is worthwhile in this new algebraic object to change the usual notion a bit. If people agree to a new usage of the word division then definitions will change accordingly. None of this is pertinent to the spirit of the original question given the context under which it was asked. All of this is highly technical.
Definitions and notions change as new mathematics is created (discovered?). This happens all the time. All you have to do is convince other mathematicians to go along with it.
https://arxiv.org/pdf/0909.2088.pdf
EDIT: Regarding what you wrote in your other comment: The analogy is not apt in my opinion. It’s hard to say zero can’t exist because the nonzero…. The moment you say nonzero means it does exist. I think a better way to look at the situation is:
I have an object that is a group under a binary operation f. There is another natural binary operation on that object that operates with f in a consistent way. That operation doesn’t form a group but if I add a symbol to my set and give these rules then both operations interact in a consistent, natural way. I get a group under the new symbol with the second operation while preserving the group under the first operation minus the new symbol.
With extended complex numbers you don’t quite preserve the structures or properties that one normally wants so I’d say it isn’t true division. It is division like.
The answer given to the person who asked the original question was the correct one. You can’t do it because doing so would break consistency and that is of paramount importance when doing new things in mathematics. There are agreed upon usages of terms and symbols in mathematics. Why call something division in the true sense of the word when it breaks the conventional usage of what that term means? But, also, why invent a new symbol to denote what is analogous to division? So we abuse notation. This is done all the time. So on the one had we’ll say to calculus 1 students 1/infinity is 0 but also say infinity is not a number. Things are done for convenience but when asked, “Is this really division?” the answer is no.
Of course you can redefine all terms and say that: A circle can be squared, I just mean something different when I say circle than when you say it. But why do that? All of this is my opinion. You disagree and that is ok.