Suppose an object (or a concept or ...) is named after the person X, in honor of Mr. or Mrs. X in mathematics: X-ian objects/ X-ic objects/ X objects.

It is natural for me to write the first letter of his/ her name in English in capital letters when referring to that object/ concept. For instance: Gaussian curvature, Newtonian mechanics, Riemannian geometry, Archimedean valuation, Eulerian graph, Platonic solids, Jacobian matrix, Noetherian rings, Artinian rings, Hamiltonian path, Hermitian matrix, Hessian matrix, ...

But why do we usually write abelian group instead of Abelian group? Or why do we usually write abelian variety instead of Abelian variety?

My strong suspicion is that perhaps the importance of abelian groups has reached us through mathematicians in languages other than English. But since I do not have a background in mathematical history, it is very likely that my guess is wrong.

Is this related to the French school of mathematics? (I know that "variety" is the French equivalent of "Manifold".) If yes, then why do we write Galois extensions?

Are there any exceptions other than abelian groups?

Also, I do not know what is the suitable tag for my question.

My answer:

I thought that perhaps the word abelian had entered the "ordinary human language", and had been accepted as an adjective. (Why did I use the term "accepted"? Because without losing accuracy, we can easily translate this word into "ordinary human language". In some sense, it has replaced the word commutative in some areas of the mathematical literature.)

That's why I checked ablian's definition from Merriam-Webster's dictionary, and this hypothesis was reinforced for me. (It states that: "adjective, often capitalized") Also, the status of the word euclidean is the same as the status of the word abelian. (It is mentioned again that: adjective, often capitalized) Also, see platonic.

When I came here to add my observations to my questions, I saw Professor K. Conrad's comment, in which he introduced an MO post (Why is “abelian” infrequently capitalized?), which is exactly the subject. I saw the answers there, and the discussions there were very convincing to me.

  • $\begingroup$ I don't know the answer to this, but interestingly enough this morning I was writing a bibliographic entry in one of my manuscripts I tinker with, and the paper's title included "abelian group". However, the photocopy of the paper I had wrote the title in all capital letters (e.g. $\ldots$ ABELIAN GROUP $\ldots),$ and my policy is to italicize titles, capitalize the first word, then only capitalize any later words if they would nearly always be capitalized. I wasn't sure whether to capitalize abelian, so I looked at some bibliographies in my books and searched online before deciding on 'a'. $\endgroup$ – Dave L Renfro Dec 27 '20 at 18:00
  • $\begingroup$ Random thought: It's standard practice for SI units that when a name is used for a unit, it's always lowercase when written out. E.g. amperes, farads, henrys, coulombs. Maybe a similar convention was popular when abelian was coined. $\endgroup$ – Sam Gallagher Dec 27 '20 at 19:01
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    $\begingroup$ Some people make up claims like "it's the greatest honor for your name to be lower case as a label" as a way of explaining the small a in abelian, but that strikes me as plain silly. Anyway, your question has been discussed before at mathoverflow.net/questions/44946/…. $\endgroup$ – KCd Dec 27 '20 at 21:39
  • $\begingroup$ @KCd Yeah, if that were the case, Galois theory would have a small 'g'. $\endgroup$ – Spencer Dec 28 '20 at 11:34

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