On the top-level tag ra.rings-and-algebras

I'm confused about the scope of the The problem is related to the ambiguity of the word "algebra", which essentially has two (rough) meanings

1. the meaning "ring", in its various meanings where one has one addition and one multiplication with some reasonable axioms, and possible elaborations and over-structures, notably coming from operator algebras or homological algebra,

2. the meaning of algebra in the more general framework of universal algebra, including, for instance, semigroups (explicitly mentioned in the current wiki of the tag).

The current wiki for basically says that both interpretations are correct, which makes this top-level tag somewhat clumsy.

Each meaning has its own interest and is related to other tags: the meaning (1) is closely related to the more specific tags: , , , , , , just to mention the more used. This is not a problem, they can be used in combination (although is unnecessary for most question of commutative algebra, since the latter is a quite well-defined area).

The meaning (2) is closely related to , to the deprecated tag , and not unrelated to .

The main problem, in my opinion, is the discrepancy between the current wiki of and its usage. Namely, looking at the last 200 questions tagged with , only about 15 of them pertain to (2), and among these 15, about 11 also pertain to (1) (typically, around embeddings of semigroups in multiplicative semigroups of rings). Among this 15, a single one is not about semigroups.

In the same period (the last 200 questions of mean since May of 2017, until now, February 2018), there were 90 questions tagged at least one of ,

Although these numbers are subject to a few minor errors or subjective interpretations, it gives the conclusion that the "universal algebra" and "semigroup" aspects, in the use of are completely drown.

In consequence, I'm inclined to remove universal algebra and semigroups from , and to give better guidelines to people who want to ask questions pertaining to (2).

Note: the role of is somewhat unclear too: if we have question about a specific structure which does not fit into usual structures with tags (groups, rings, etc, or close variants), one of its problems is that people who have questions about this specific structure may simply ignore the notion of universal algebra, and also the wiki "study of algebraic structures and properties applying to large classes of such structures" is not really encouraging.

• Here is link to a related discussion in chat. Todd Trimble explicitly mentioned that he plans to talk about this tag with François G. Dorais - I am not sure whether he got around to that. – Martin Sleziak Feb 18 '18 at 15:50
• My opinion is that it's a bad idea to use the rings-and-algebras tag if "algebras" is meant in more of a universal algebra sense. Thanks for the reminder to look into this and hopefully fix it. – Todd Trimble Feb 18 '18 at 16:04
• I, too, think this is a mistake, but it is probably one inherited from the arxiv. There, the rings and algebras tag description is identical to the one used on MathOverflow, and must predate the MathOverflow tag by almost 20 years. Unfortunately, on the arxiv, there are no alternative tags for universal algebra, lattice theory, semigroup theory or linear algebra. – Keith Kearnes Feb 18 '18 at 18:50
• @KeithKearnes is right. That tag was inherited from the arxiv, just like all other xx.* tags. – François G. Dorais Feb 18 '18 at 18:58
• @ToddTrimble I think the key is the 's'. To me algebras are a type of ring, but algebra is a field of study (which could be described as universal algebra and its applications). – François G. Dorais Feb 18 '18 at 19:05
• @FrançoisG.Dorais "algebras" is also used in universal algebra unrelated to rings, whence the ambiguity. – YCor Feb 18 '18 at 19:07
• @YCor context matters: what I wrote is what I assume is meant when someone I just randomly met says "I study algebras" vs "I study algebra". A lot of math words are used in a variety of incompatible ways. Some are hopelessly ambiguous out of context (e.g. "lattice" in universal algebra vs number theory, "field" in algebra vs geometry). To me, "algebras" has a primary meaning and a secondary meaning, but it is indeed ambiguous. – François G. Dorais Feb 18 '18 at 19:19
• @FrançoisG.Dorais I perfectly agree. I just mean that this extra-use is part of the origin of the problem. From the end of my post it's perfectly clear that I'm suggesting to explicitly remove this non-ring meaning of "algebra(s)" from the use of the tag "rings-and-algebras". – YCor Feb 18 '18 at 19:36
• Maybe this is for separate discussion/separate post on meta, but if some of the areas currently mentioned in the tag-excerpt are going to removed, it is natural to ask which top-level tag should be used for them. (Since several posts here on meta recommend that every question should have top-level tag.) – Martin Sleziak Feb 19 '18 at 5:44
• @MartinSleziak I think the consensus that the non-ring stuff (2) is removed is a necessary first step, and that breaking the "rule" of top-level tags is less a problem than the current problem. A second step is a serious thinking on the use of tags related to (2). It requires discussion, but especially a serious work of detecting what is the state of art (what are the relevant tags, what are obvious merge - the most obvious example at this level would be semigroups/monoids); I'll be happy to spend hours on this at some point. – YCor Feb 19 '18 at 8:16
• I think it is best to leave it with the same meaning as ArXiv since probably people will assume it has the same meaning anyway without reading carefully. The ArXiv heading is not good, for instance, I never know whether to post a semigroup paper as group theory because in MSC it goes under groups and its generalizations, or under ra.rings-and-algebras because that is what ArXiv says to do. But unless we change simultaneously with ArXiv I think it is best to stay consistent with them. – Benjamin Steinberg Feb 21 '18 at 21:02
• @BenjaminSteinberg "since probably people will assume it has the same meaning anyway without reading carefully" I have substantiated the opposite claim with statistics. – YCor Feb 21 '18 at 22:28
• @YCor, What do the universal-algebra tagged entries use as a top level tag then? – Benjamin Steinberg Feb 21 '18 at 22:59
• @BenjaminSteinberg Including set-theory as top level tag, on the last 100 post tagged universal-algebra (goes back to July 27 14): NONE (27), ct.category (23), set-theory (15), gr.group-theory (14), ra.rings-and-algebras (14), lo.logic (12), ac 4, co 3, ag 3, and a few more. 13 of these had at at least 2 top-level tags. Removing set-theory from the list just increases the NONE list. Above is subject to minor errors. – YCor Feb 22 '18 at 0:31
• @BenjaminSteinberg I am not exactly sure whether your question is which top-level tags should be used with (universal-algebra) or which ones are currently being used. In any case, I have tried to give some kind of current usage summary here in chat. – Martin Sleziak Apr 2 '18 at 0:38

How synchronized/copacetic/friendly do we want MathOverflow and ArXiv to be?

I can put up an argument for keeping the status quo (new users will identify the two tag systems, researchers in structure theory will look under the ra tag as well as other pertinent tags, other users will use the ArXiv tag in their questions on a paper), but I'm a status quo kind of guy, and that can be viewed as impeding progress.

I actually see rings as term for several classes of structures that include near-rings, lattices, and systems with multiple operations that are usually binary. So the view I have is different from Ycor's, and I am not seeing the problem as he sees it.

I think the issue isn't regulation of tag use or tag description, so much as how do we want a "MathOverflow classification" system to follow or differ from other classification systems like MSC or ArXiv. That is too big a subject for me to tackle, and I am not familiar with Ycor's (1), so let me narrow the scope of the question to "How do we handle tagging posts geared toward (2)?".

I see two terms that are appropriate here: 'general algebra' and 'structure theory'. Granted there are overlaps with other fields like category theory, model theory, and other parts of mathematical logic. How about for those questions with four or fewer tags, those deserving of one of the above terms gets that term, and the remaining few are handled on a case by case basis? If not this solution, what would the current generation of students and researchers of (2) like to see happen with MathOverflow that might work for their generation and a few more to come?

Gerhard "Yes, I'm Thinking System Design" Paseman, 2018.02.19.

• I at least agree with you that the question "How do we handle tagging posts geared toward (2)?" is the main one. Opting for status quo means: considering ra.rings-and-algebra as the relevant tag, which will result in continuation of the current situation: only a small fraction of questions pertaining to (2) will be effectively posted in ra.rings-and-algebra (drown in a large 90% of questions arounds algebras in the sense (1)); others will be somewhat randomly dispatched into other tags: semigroups, monoids, universal-algebra, depreciated abstract-algebra, and others. – YCor Feb 20 '18 at 15:50