Comment by yellowcake0

Calling it addition is hairy here. Do you just mean an operator? If so, I'm with you. But normally people are expecting addition to have the full abelian group properties, which this certainly doesn't. It's not a ring because it doesn't have the multiplication structure. But it also isn't even a monoid[0] since, as we just discussed, it doesn't have associativity nor unitality.

There is far less structure here than you are assuming, and that's the underlying problem. There is local structure and so the addition operation will work as expected when operating on close neighbors, but this does greatly limit the utility.

And if you aren't aware of the terms I'm using here I think you should be extra careful. It highlights that you are making assumptions that you weren't aware were even assumptions (an unknown unknown just became a known unknown). I understand that this is an easy mistake to make since most people are not familiar with these concepts (including many in the ML world), but this is also why you need to be careful. Because even those that do are probably not going to drop these terms when discussing with anyone except other experts as there's no expectation that others will understand them.

[0] https://ncatlab.org/nlab/show/monoid