What is the tag [derivations] for?

Since the tag has no usage guidance at the moment, I do not know when it should be used, which is unfortunate.

• My old algebra memory says: a derivation on a ring $R$ is a map $D : R \to R$ satisfying $D(a+b) = D(a) + D(b)$ and $D(ab) = aD(b)+D(a)b$. Perhaps that is what the tag is for. Commented Jun 10, 2023 at 13:04
• @GeraldEdgar My concern is that some people might use it for questions on derivatives, i.e., as a synonym for differentiation Commented Jun 10, 2023 at 13:06
• I could also see it getting used for questions about logical derivations, i.e. formal proofs in the sense of natural deduction or similar systems. Commented Jun 10, 2023 at 14:42
• Some stats related to the tag (derivations) can be found in the MO editors' lounge. Tangentially, this tag now has a tag-excerpt. Commented Jun 28, 2023 at 9:45

I think that indeed:

• Perhaps differential-field would work for most algebraic uses? (I'm no expert but these are essentially the only algebraic derivations I personally have worked with. I understand that the concept is more general but we don't need a tag for every concept.) Commented Jun 20, 2023 at 2:04
• Actually, I just learned from Wikipedia that differential-algebra might be both more specific and comprehensive for the algebraic uses of the term. Commented Jun 20, 2023 at 2:10
• Re: differential-calculus I agree that it's not really "research-level" but isn't that a little disparaging towards Newton, Leibniz, Gauss, Euler, Bernoulli, Bernoulli, Fourier, Bernoulli (is there a fourth Bernoulli?), Jacobi, Riemann, etc.? Commented Jun 20, 2023 at 2:19
• @FrançoisG.Dorais Differential field is a field endowed with a derivation. This does not cover "derivation" in several respects: a derivation can be defined on a ring $R$ (not necessarily associative), and can also be defined from $R$ into an $R$-module. Also, the set $\mathrm{Der}(R)$ of derivations on $R$ forms a Lie algebra and can be considered as a whole without singling out a single derivation. In important cases this is the Lie algebra of the group $\mathrm{Aut}(R)$, and occurs quite naturally. Quite many of the questions now with the tag would not be tagged "differential-algebra".
– YCor
Commented Jun 20, 2023 at 3:41
• Yes, I am aware of the existence of such generalizations. I also believe that they are useful in contexts that I am not familiar with (or perhaps even some familiar contexts that can't remember at the moment). But this is mathematics as a whole, not actual MO usage. For tagging purposes, the tag name should ideally be both useful (in the MO usage sense) and unambiguous (when possible). The tag derivations is neither! Commented Jun 20, 2023 at 4:12
• @FrançoisG.Dorais in any case, I agree that the tag differential-algebra is better defined and should be used more often. It has no usage guidance. A draft: "Fields or rings endowed with one of several derivations. Differential Galois theory, algebraic study of ODEs/PDEs, etc."
– YCor
Commented Jun 20, 2023 at 7:22

Another possibility is removing the tag altogether. Let us consider the pros and cons:

• I can easily see people repeating the same mistake and misusing the tag, even with better usage guidance. So keeping the tag tidy will require further effort.

• There seem to be only few questions that would keep the tag after the retagging YCor proposes, and it seems to me that they all contain the word "derivation" in the title. So it will be simple to search for questions on derivations even without the tag.

• This may be a case where it is better to have a small amount of narrower tags than a tag for the overarching concept. Commented Jun 20, 2023 at 2:07