Talk:Lattice
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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.
- 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.
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 Rn or Cn 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)