What is the homogeneous-elements tag supposed to be used for?

Consider the tag .

I ran into this tag today used to describe the rate parameter of an exponential distribution. Given my experience in probability, I'm pretty sure this isn't standard terminology.

Algebra is not my specialty, but it looks like, besides this, this tag is used to describe "homogeneous ideals," "homogeneous polynomials," etc. There is no guidance in the tag wiki which describes what this tag is.

Could someone who has more expertise than I do suggest what to do about this tag, and if we decide to keep it, could someone please put a description in its tag wiki? For now, I'm going to remove this tag from the aforementioned probability question.

• AFAICT the question where the tag was created and the questions added soon after that were from commutative algebra. I have pinged the tag-creator, maybe they join the discussion here on meta and explain the intended usage (or even suggest the tag-wiki). – Martin Sleziak Dec 19 '17 at 16:03
• @MartinSleziak Thanks for pinging me. I feel connected and stuff. – StudySmarterNotHarder Dec 19 '17 at 19:48

This tag appears to be intended for questions about homogeneous elements of graded algebraic structures such as graded rings and graded modules. However, as such, the tag seems wholly redundant with the already-existing tags , , and . So I would propose that the tag be removed and that questions with it be retagged with a subset of those tags as appropriate.

• It seems that the result of this discussion is clearly in favor of removal. I have already removed the tag from a few questions - only 5 questions remain. It would probably better if somebody knowledgeable in commutative algebra, graded rings, etc. had a look at the remaining ones - so that appropriate tags can be added at the same time (since the questions are going to be bumped by removal of this tag anyway). – Martin Sleziak Jan 1 '18 at 15:30
• I've retagged the last 5. – Eric Wofsey Jan 1 '18 at 17:21
• Thanks for doing that. The tag is now removed also from the system. – Martin Sleziak Jan 2 '18 at 9:10

If you google "homogeneous elements in algebra". You'll get many hits. Here's one on Wikipedia:

Of course they're referring to homogenous polynomials which of course is way more common. Can anyone reason why they use the "elements" form instead?

Anyway, googling the same ".. in statistics" or ".. in probability" turns up virtually nothing AFAICT.

• Probably because it makes sense beyond polynomials. For example, tensors form a graded algebra, and the rank-$n$ tensors are the homogeneous elements of the tensor algebra, yet nobody would normally consider e.g. a rank-2 tensor a polynomial of degree 2. – celtschk Dec 19 '17 at 22:57
• For everything beyond polynomials, though, the graded-rings or graded-modules or graded-algebras tags would do just as well; the polynomial case is the only one that people unfamiliar with general graded whatevers are likely to need. – Misha Lavrov Dec 20 '17 at 0:25
• @MishaLavrov Do you think creating (homogeneous-polynomials) and adding a synonym between this new tag and (homogeneous-elements) would be a reasonable way to go. (In this way, we would have a single tag which encompasses both the more frequent usage and the more general one.) Or. if the tag for homogeneous polynomials is created, would it be better to keep the two tags separate? – Martin Sleziak Dec 20 '17 at 8:09
• @MartinSleziak If a homogeneous-polynomials tag exists for the specific usage (and graded-algebras exists for the general) then I don't see a reason to keep homogeneous-elements around at all, unless this makes the transition process more convenient somehow. – Misha Lavrov Dec 20 '17 at 13:32
• @MartinSleziak: I don't see any reason for either of those tags to exist. They are almost completely redundant with graded-rings, graded-algebras, and graded-modules. To the extent that homogeneous polynomials are a topic of interest in a less ring-theoretic setting, polynomials should do just fine for them. (Incidentally, graded-algebras and graded-rings should probably be made synonyms, though I'm not sure which should be the master.) – Eric Wofsey Dec 20 '17 at 20:31
• Actually, I might propose making a single tag graded-algebra for all graded algebraic objects and making graded-rings, graded-modules, and graded-algebras all synonyms of it. – Eric Wofsey Dec 20 '17 at 20:39
• Since two users brought up some arguments why the tag (homogeneous-elements) should be removed, maybe it would be nice to post this as a separate answer. (So that it is more visible and we can get more feedback on this - by votes from users on that answer and possibly also by votes.) Would you be willing post an answer @MishaLavrov... – Martin Sleziak Dec 21 '17 at 12:28
• ...or @EricWofsey? I will add that a good place where to discuss the suggestion about various graded-stuff-related tags would be the tag management thread. (Although as 2017 ends soon, maybe it would make sense to wait for a new edition.) – Martin Sleziak Dec 21 '17 at 12:30