# Talk:Lattice (mathematics)

The first definition has been standard at least since the 1930s and probably since Dedekind worked on lattice theory in the 19th century; though he may not have used that name. -- JanHidders

The definition is given as:

- A
*least upperbound*of*V*is an element*x*in*L*such that- for all
*y*in*V*it holds that*y*<=*x*, and - for all
*z*in*L*it holds that if*z*<=*v*for all*v*in*V*then*x*<=*z*.

- for all
- A
*greatest lowerbound*of*V*is an element*x*in*S*such that- for all
*y*in*V*it holds that*x*<=*y*, and - for all
*z*in*L*it holds that if*v*<=*z*for all*v*in*V*then*z*<=*x*.

- for all

Isn't the first inequality in the second subbullet under both of the main bullets backwards? Shouldn't it be *v* <= *z* in the first case and *z* <= *v* in the second case, rather than vice versa?

*Yup.*

Aren't finitely generated subgroups of **R**^{n} or **C**^{n} also called lattices? I wonder if they are related to the order-lattices. --AxelBoldt

Discrete subgroups, rather than finitely-generated subgroups, I think. E.g., <1,π> is a finitely generated subgroup of **R**, but it isn't a lattice. They aren't related to the type of lattice described in the current article. I was going to add a mention of them yesterday, but I couldn't think of anything much to write.

Zundark, 2001-08-20

I see. Maybe Minkowski's theorem about the number of lattice points in a convex set could be linked. --AxelBoldt The new material science definition seems to be the same as a discrete subgroup. --AxelBoldt

Yes. I think what we should do is to add the discrete subgroup definition, and then modify the materials science definition to mention that this is a special case of one of the mathematical definitions. --Zundark, 2001-08-21

I added some basic material to the order-theoretic side of lattices (filters, ideals, join-irreducibility, ...). I also tried to improve the structure of the page by giving some subsection-headings and moving the category/morphism part upwards (it did not seem to fit elsewhere and is quite basic anyway).

Considering the huge size of the lattice theory part, it might be a good thing to have a disambiguation page for the four notions in this article and put up a new page called *lattice theory* or *lattice (order theory)*. -- Markus (no login), 2003-09-22

- Yes, we probably should split the page. We need to change all the links first, however. We already have a redirect called lattice theory, so we can start changing links to point to it. I was going to start doing this, but I think I should wait to make sure that everyone agrees that "lattice theory" is the best name for the new article (because I don't want to have to change them all a second time). --Zundark 22:18, 22 Sep 2003 (UTC)

Isn't the image a little large for a disambiguation page? (As sweet a picture as it is.)

--cfp 01:51, 23 Nov 2004 (UTC)

## Xmas?[edit]

What is Xmas doing on the Lattice page? Jon Awbrey 15:54, 29 December 2005 (UTC)

## Page organization[edit]

It looks like we have three competing alternatives here:

- A page with paragraphs describing some different meanings of the word (proposed by Patrick)
- A disambiguation page with an introductory paragraph and definition (proposed by Michael Hardy)
- A disambiguation page with a minimal introductory line (proposed by Smack)

Out of these three options, the last one conforms the best to MoS:DP. However, Michael Hardy and Patrick think that we need more explanatory content. Let's talk. --Smack (talk) 06:50, 3 March 2006 (UTC)

- Why would anybody be against explaining the relationship between various meanings? For such a page MoS:DP does not apply, a page with content is not a disambiguation page.--Patrick 08:56, 3 March 2006 (UTC)

- See also Wikipedia_talk:Manual_of_Style_(disambiguation_pages)#Page_with_content_is_not_a_disambiguation_page.--Patrick 09:01, 3 March 2006 (UTC)

Suppose there's a "trunk" disambiguation page (oh--I see--there is!). It could say there are elephants' trunks and trunks that carry luggage. Those meanings seem unrelated. But with lattice, there's one **basic** meaning that **explains** the others. Where in Wikipedia, if not on this page, would that fact get explained? Michael Hardy 19:22, 3 March 2006 (UTC)

- It seems that the concept that Patrick is looking for is what has been proposed as a Wikipedia:Root page; personally, I think a separate label for such pages is unnecessary, but I do like the idea. I would favor
**removing**the {{disambig}} tag from this page and instead strengthening the content that explains the relationships among the various technical usages. --Russ Blau (talk) 14:16, 4 March 2006 (UTC)

- I know that a page with content is not a disambig. However, you can't use that as an argument in favor of including more content - that would be circular logic. If we leave it the way it is, it will not be a disambig, and if we cut out all of the content again, it will be. That doesn't answer the question of whether or not it
*should*be a disambig.

- I know that a page with content is not a disambig. However, you can't use that as an argument in favor of including more content - that would be circular logic. If we leave it the way it is, it will not be a disambig, and if we cut out all of the content again, it will be. That doesn't answer the question of whether or not it

- I think that the answer to that question is simple: there are many ambiguous incoming links, searches, etc., and they need to be disambigged. This is all laid out in detail on Wikipedia:Disambiguation. Furthermore, if we're going to disambig, we should follow the rules laid out in MoS:DP. As far as root pages go, they're not policy yet, and IMHO they have some issues (see Wikipedia talk:Root page#Roots vs. dabs). --Smack (talk) 06:46, 6 March 2006 (UTC)

I can see the argument for many of the meanings being related to each other. However, since there are also other links on this page that do not relate to the mathematical meaning of lattice, there is need for this to be a disambiguation page. However, one option would be to create a new article solely about mathematical lattices, and take all the lattice-related information from this page and put it on there (and clean it up to seem like an article). Then there could be a separate disambiguation page linking to that new article and the other lattice-related links. Thoughts? -- Natalya 12:30, 15 March 2006 (UTC)

- I agree with Nataly's view. The fact that there other articles like lattice (order) and lattice (group) supports this view. So I suggest that any conent to be removed to those articles or a new article perhaps lattice (maths) to be created and all relevant content to be mreged into it. Mahanchian 20:41, 16 March 2006 (UTC)