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