# Two tags for partially ordered sets

There are currently two tags for partially ordered sets: posets and order-theory. In my opinion, they mean the same thing. I see three posibilities.

1. Merge the tags.
2. Make them synonymous.
3. Keep them as they are.

I think 2. would be the best option. What do you think?

• Even though I am the top user in order-theory (mathoverflow.net/tags/order-theory/topusers), could someone explain to me what is "order theory"? I don't usually use this term. Is it widely used in some areas? Sep 15, 2013 at 0:41
• At least one of the posts with label order-theory is about something called "cyclically ordered sets". Sep 15, 2013 at 13:40
• @Joel, see e.g. Gierz, Hofmann, Keimel, Mislove, and Scott, "Continuous Lattices and Domains", 2003. Although one might consider these questions to be about partial orders (or similar structures) I feel "order theory" is more common for the topics discussed in the book. Using "partial orders" to refer to all of these topics feels like using "groups" to refer to "group theory" and the theory of any structure build upon groups. Sep 15, 2013 at 22:03
• Joel, I'm genuinely surprised at your puzzlement. Order theory is, naturally, the theory of order(s) -- e.g. from where I'm sitting now, I can stretch out my arm and pick up Davey and Priestley's undergraduate textbook Introduction to Lattices and Order. People in different areas use the word "order" in slightly different ways (just as people in different areas use the word "ring" in slightly different ways, but "ring theory" is still a useful term). Presumably most mean partial order, but some might mean total order, preorder, etc.; they're all bound up together. Oct 2, 2013 at 22:34
• @Tom, of course I am very familiar with that theory; I just don't often hear it described as "order theory". But perusing google with that term, it seems that the explanation is that this usage of "order theory" is more common among category theorists. Oct 3, 2013 at 13:51

I am involved with questions having these tags quite a lot, but my preference would be to merge them both into a tag called "partial-orders".

I rarely use the term "posets" myself, and I would usually not describe the topic as "order-theory", as opposed to the theory of partial orders. Most set theory books do not define anything called an "order", but rather define various kinds of orders, such as partial orders, linear orders and so on. I imagine that one could find some definitions of order to mean what I would call a partial order, but I think this is less common.

So, I propose that we merge these tags into: partial-orders.

Many of the questions currrently in order-theory concern exclusively linear orders, and these questions should be tagged with a tag called "linear-orders".

I think this has been discussed before. The problem is that "order theory" is sometimes mistakingly construed as the theory of orders in the sense of ring theory and the theory of orders in the sense of group theory. These are rare but this kind of confusion is problematic with tag synonyms.

• Now, with tag-wikis this problem could be addressed by writing one. Of course, somebody might still not read it and use a wrong tag, but then there are many reasons why people use wrong tags.
– user9072
Sep 14, 2013 at 22:30

I disagree. There are questions about posets that wouldn't really fall under order theory, at least as I understand the term. For instance, my question Posets of finite sequences are highly connected is about the topology of a certain poset.

• I thought the trend in comments was opposite to how Andy Putnam takes it. I thought people were saying order theory is better for question relating order to other structures, and poset was better for more strictly poset questions. That view persuaded me, but perhaps I misunderstood. I don't think I have used either tag or particularly looked for either. Sep 18, 2013 at 11:57
• @Colin McLarty : My last name is spelled PutMAN... Sep 19, 2013 at 19:28

While order theory often deals with posets, there are weaker relations such as preorders as well as relations compatible with orders that are studied in order theory. I recommend the tag poset when the emphasis is on the structure, and order theory when the emphasis is on things "related to" the relation. There will be overlap, but that is unavoidable.

It does not seem as if "posets" and "order-theory" are entirely synonymous since the tag order-theory seems to encompass more questions than the tag posets. These tags should be kept as they are.

For example, a question about Heyting algebras, frames, or Boolean algebras could justifiably be tagged as order-theory. On the other hand, for several reasons, it is harder to see questions about Heyting algebras, frames, or Boolean algebras tagged as posets since Heyting algebras, frames, and Boolean algebras are very specific types of posets. Furthermore, structures such as Heyting algebras, lattices, and Boolean algebras can and usually are defined in terms of their lattice operations instead of their partial ordering. Therefore for these structures, the tag "order-theory" will probably be more appropriate than the tag "posets".

I should also mention that there are structures similar to partial orders which are not quite partial orders. For example, as mentioned before, preorders, and cyclic orders fall under "order-theory" but these objects are not quite partial orders. Also, sometimes partial orders are endowed with extra structure and these objects should not be seen as simply partial orders (i.e. partially ordered topological spaces).

• I would think that a use of the heyting-algebras, lattices or boolean-algebras tags would be better placed for such questions than order-theory. Oct 1, 2013 at 23:17
• For a question about Heyting algebras or lattices, I would probably include both a heyting-algebras or lattice-theory tag and an order-theory tag. Dec 3, 2013 at 17:09