# What "lattices" do we need?

The tag is a problem since a long time. Over the years there were varied discussions about this, but no clear resolution emerged, though some things were done to better the situation. This is an attempt at a fresh start of the discussion to find a long-term solution.

## The current situation

The tag is relatively large (at 300+ questions) its (highly-visible) tag-wiki excerpt says:

Lattices as they are used in number theory. (Not to be confused with lattice theory or lattices as used in physics!)

But, its less visible tag-wiki says:

This tag is ambiguous, so we recommend against its use. See https://meta.mathoverflow.net/a/1761/121

Questions with this tag tend to be about either partially ordered sets, or discrete subgroups of Lie groups, or finite rank free abelian groups equipped with a quadratic form.

The tag at 100+ questions is described as:

The theory of lattices in the sense of order theory. For the number-theoretic notion, use the tag "lattices" instead.

And mainly it is actually used only for this. (Thus, this is fine.)
There is also tag , which is more specialized.

## Problem

The main problem as I see is that the tag is half-way deprecated while there is no clear alternative except for the order theory questions (and those about euclidean lattices).
[A side-problem is that at one place a tag is recommended that is advised against elsewhere but that is easy enough to fix.]

Personally, I am not even quite clear about all the exact notions users might like to tag with some variant of "lattices," and in earlier discussions sometimes users were talking sideways for reasons like this. For example while I can see what is roughly meant with "Lattices as they are used in number theory." it is not exactly clear either.

## The immediate aim

What I would like to achieve is to get some list of notions/subjects that might fall under "lattices" each with a viable name for a tag. (For the order theory context this seems essentially achieved, via , but perhaps somebody sees room for improvement. Added: It appears not everybody is happy with the name of this tag. In fact, I do not consider it as optimal either as it does not directly convey what that it is about the order context. A solution could be to combine it with a related notion, like in . Would there be any?)

Not each answer needs to, perhaps even should not, try to hash out a complete solution (we tried this several times already), but rather I envision that somebody writes down so to say "their kind of lattices", that is, what "lattice" should encompass in one given context. And, if this notion should get a tag, how this could be named distinctively.

Based on such a list we could then create the new tags, definitely deprecate and over time distribute the legacy questions as needed over the new tags.

## tl;dr

Describe for one given context what "lattice" should encompass and propose a tag-name for this.

• I see room for improvement. Probably there is no way to achieve a separation between the number-theoretic notion and the order-theoretic notion, since both are concerned with "lattices", and so there will always be new questions posted using that tag with different senses. When I study what I call lattices, it is almost never with the number-theoretic sense. I don't take myself to be doing "lattice-theory", but rather just studying lattices. Since I expect most other researchers feel similarly, the tag will inevitably have questions mixing the senses of different kinds of lattices. Commented Mar 13, 2016 at 2:10
• @JoelDavidHamkins thank you for the input. We managed to do it for "divisors", "sequences" and some similar cases. I do not see why we should not be able to do it for "lattices." (It seems for once I am the optimist among us two. :-) ) Part of the solution there was to combine the term with some related notion that sets the context: in sequences-and-series it seems clearer what sequence is supposed to mean, likewise in divisors-and-multiples. Would there be such a notion for lattices in the order context, say some dual or more general notion?
– user9072
Commented Mar 13, 2016 at 12:02
• On math.SE there are currently three big "lattice" tags, [lattice-orders], [integer-lattices] and [vector-lattices].
– Asaf Karagila Mod
Commented Mar 13, 2016 at 18:12
• @Asaf thanks for this useful information. I just note there is also [lattices-in-lie-groups], which I find a very good tag-name for the . "discrete subgroups of Lie groups."
– user9072
Commented Mar 13, 2016 at 18:22
• Well yeah, but that tag is quite small so I didn't think to mention it here.
– Asaf Karagila Mod
Commented Mar 13, 2016 at 18:25
• To those with 10k on MSE, this link is of interest.
– Asaf Karagila Mod
Commented Mar 13, 2016 at 18:27
• Since @Asaf's link it may be of interest also to those who don't have 10k on MSE, let me post a screenshot. I don't believe there's anything there that should not be publicly disclosed; the post was apparently deleted just "to de-clutter" the thread. Commented Mar 15, 2016 at 9:50
• Now that Stack Exchange has tag warnings, I think that we should use them for deprecated tags. Commented Mar 18, 2016 at 13:59
• @FedericoPoloni this is an interesting idea. You might want to raise it separately.
– user9072
Commented Mar 18, 2016 at 14:09
• Would people interested in order-theoretic lattices be ok with a tag [lattices-and-posets]? I think that [euclidean-lattices] and [lattices-in-lie-groups] are very good options for the other three meanings. If tag warnings work well, flagging [lattices] and adopting the three separate tags could be a good solution. Commented Oct 19, 2021 at 13:08