# Should there be a separate tag for partially ordered sets?

Users who are on this site some time might have noticed that a separate tag for posets has been created and removed a few times in the past.1

Certainly, partial orders are quite an important topic. On the other hand, there already is the tag which is suitable for questions about partial orders.2 So by creating one additional tag we would not gain too much and we would use already two of the five possible slots for tags on such questions - since they would be typically tagged by this specific tag and also by (order-theory).

So my main questions are:

1. Should there be a separate tag for partial orders? Would such tag be actually useful?
2. If the consensus is that we do not want a separate tag for this purpose, would creating synonyms $\to$ , $\to$ , etc., be useful?

The intention of the synonyms suggested in the second point is that new users might be unaware that is the tag suitable for questions about partial orders, so they might be looking for a tag with "poset" or "partial order" in the name. (As witnessed by repeated creation of such tags in the past.)3

If we compare the situation with MathOverflow, they have separate tags posets (198 questions) and order-theory (378 questions). If we look at those tags, there are 87 questions which have both tags and it seems that there is a fair share of questions which are about partial orders but do not have (posets) tag. So I am not sure whether separation into two tags contributes to having the site better organized on that site. This was discussed briefly also on Mathoverflow Meta, although the discussion there seems to be inconclusive: Two tags for partially ordered sets.4 (And, needless to say, the two sites are not exactly the same - so not everything that works there would also work here and vice versa.)

I would suggest to keep this discussion specifically about partial orders. I do not know about attempts to create a tags for linear orders (total orders) in the past - but if the need arises we can start a separate discussion about such tag. There already exists a separate discussion about well-orderings - and moreover well-orders seem to me sufficiently different from partial orders, so it's probably better not to discuss them in the same topic.

1For example, the tag was created recently in December 2017 - including tag-excerpt and tag-info. Shortly after removal by a moderator, the tag reappeared in January 2018 - including a new tag-excerpt and tag-info. Using the query from this post you can find several past occurrences of posets, poset and partial order. For example, the tag was created and removed in 2013, 2014, 2015 and 2018.

2The current revision of the tag-excerpt says: "Order theory deals with properties of orders, usually partial orders or quasi orders but not only those. Questions about properties of orders, general or particular, may fit into this category, as well as questions about properties of subsets and elements of an ordered set."

3To some extent in the spirit of this older discussion: Tags as redirects for correct tags. (Although the idea suggested there seems to be abandoned.)

4Looking at that discussion reminded me that there is a possibility that some users might mistakenly use this for order of a group. (Typically if a user does not read the tag-excerpt and decides only based on the name of the tag.) If you look at the 75 questions tagged order-theory+group-theory, many of them are mistagged. However, this is only tangentially related to this discussion.

• I do think a synonym posets $\to$ order-theory (among others) would help users tag questions in this instance. As an aside, “order theory” sounds kind of fancy. What I mean is that I’ve used posets fairly frequently, but I don’t think I’ve ever proven anything about posets the same way I’ve proven things about groups, for example. So, a posets tag feels like a better fit for how I use posets than order-theory, in my mind. However, this opens a “tool vs. central object of study” distinction that’s not present for other tags, as far as I know... – pjs36 Jan 23 '18 at 2:43
• @pjs36 I am not sure whether these are exactly examples similar to what you are asking about but (rings) and (ring-morphisms) are synonyms of ring-theory and (algebraic-closure) is a synonym of field-theory. – Martin Sleziak Jan 23 '18 at 16:09
• I'm not entirely sure that we should take a lot from the tagging system on MathOverflow. It has some merits (e.g. the top-level arXiv tags, but even those are eroding), but for more specific topics it's more haphazard, and as the site is significantly smaller and less active, this is not a terrible problem. – Asaf Karagila Jan 26 '18 at 10:25
• @AsafKaragila Yes, I am aware that MO and Math.SE are different in many aspects, including some things related to tagging. (I have included this also in the question.) Still it is at least some kind of data point - when somebody considers whether or not this tag could be useful, checking how the same tag is doing on MO cannot hurt. – Martin Sleziak Jan 26 '18 at 14:50
• When tags become too narrow, there is not enough new posts to make worth while to bother reading it. Order theory tag needs to be improved to weed out algebra - order of an element in a group stuff. – William Elliot Jan 27 '18 at 20:49

I guess since I've been the only one to weigh in, I'll do so "formally" in an answer.

Synonyms are pretty cheap to create, and don't have the same downsides as creating too many tags, right? I personally didn't really know "order theory" was a subject in its own right, before seeing the tag here (and thus wouldn't have searched for it / tried to apply the tag to any questions I may have had about partial orders).

• I personally think synonyms are enough, and that we don't need a separate tag for partially ordered sets.

I suspect (but have no evidence to support my suspicion) that a significant portion, if not a clear majority, of questions tagged with deal with partial orders, and not something less stringent like a preorder (although the more specialized topic of has been relatively popular, with 821 questions using the tag at the time of writing). Thus, I suspect that splitting off questions about partial orders would leave the tag significantly less used, and it's not terribly popular to begin with.

My personal feelings are that the topic is not as popular as e.g. where significant tag specialization is necessary, nor does there seem to be a need to split questions about "genuine" order theory from those of a more elementary nature (as with the splits in our or tags and their elementary versions). But, I would especially welcome dissenting opinions here.

While there is potential for misusing for, e.g., the concept of "order" in group theory, a more specialized tag like isn't a good fit for all questions pertaining to more general types of orders. I do think the current name is most appropriate for the master tag.

• It is infelicitous that "order" is a term with varied meanings (besides group theory, polynomial degrees, matrix ranks, and derivative hierarchies come to mind). Still, and noting that similar potential for misuse exists in other tags, I'm happy with picking "order theory" as the master synonym. – hardmath Feb 2 '18 at 16:57
• I created those syns but massively disagree with "and don't have the same downsides as creating too many tags, right?" as a general statement. Some synonyms are toxic, way more than any tag ever could be. The worst is it's a silent poison. Questions end up with non-sense tags and the poster might not even realize. – quid Feb 16 '18 at 0:15