Talk:Map of lattices

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I have made a graph that shows different types of lattices and their relationships.
I think it may help a student of lattice theory but I wanted to discuss it here before including it. Please tell me

  • if there are any mistakes.
  • if more powerful relationships exist (references would be nice).
  • if some types are missing, or superfluous. In particular I am not sure we should include types that have no articles (like geometric lattices) or that are somewhat exotic (MV-algebra). The current graph is certainly too big.
  • if such a graph is even useful.

Discussion of the visual aspects can happen later. Naturally the graph will be remade as a clickable map. Thanks for your comments.
Ceroklis 14:27, 26 September 2007 (UTC)

Providing references for the edges in such a large diagram would be troublesome. Providing a copy of a diagram in a published work would be referenced, but you would need to get the copyright holder's permission. The benefits of a correct diagram seem to be outweighed by the uncertainty of its correctness and the difficulty in proving correctness. Such a graph is certainly useful and a nice contribution to the world. The difficulty is whether it is an appropriate (direct) contribution to wikipedia, due to the likelihood of it either being unsourced or a copyright violation. On the other hand, it is certainly not impossible to fix, merely a lot of work. JackSchmidt 15:45, 26 September 2007 (UTC)

The lack of sources wouldn't make it much worse that most of the articles on lattice theory :) But creating a page with justification (short proof or reference) for each edge is not a problem. The question is what to put exactly in the map and how to present this information (in a sepearate page ?). Ceroklis 18:04, 27 September 2007 (UTC)

Any chance of squeezing in antimatroid above semimodular lattice somewhere? —David Eppstein 18:41, 26 September 2007 (UTC)

Not really. First antimatroids themselves are not lattices, their feasible sets are. Second, this map presents the main properties a lattice can have, not particular examples. For instance the example "power set with inclusion" is not in the map. The general category "boolean algebra" is. Ceroklis 18:04, 27 September 2007 (UTC)

By the way, I realize this is just a Hasse diagram of lattice families, but it might make some sense to attempt a concept lattice of them instead, with examples of lattices in each class. E.g., seeing the diagram above, it is natural to wonder whether there are any modular geometric lattices that are not projective; the concept lattice would answer that question. Another question: what is the relation between orthomodular lattices and modular lattices? —David Eppstein 18:30, 27 September 2007 (UTC)

A concept map would certainly be great. However it would demand an enormous amount of work, since for each possible subset of properties you would have to either prove no lattice can have these properties and not the others, or find a counterexample. You may be familiar with the book Counterexamples in Topology, in which they do exactly that for the properties of topological spaces. But it requires over 200 pages of proof! Nowadays you can use automatic theorem provers and model searchers to speed things up but it is still a lot of work, especially since you need infinite models in some cases. This would be material for a book, not a wikipedia entry. An lattice theory equivalent of the book I mentioned may exist but I don't know of any.
To go back to the current map the idea is to summarize some of the most important relations (that are proven in any introductory book), not to be exhaustive. It's not perfect but I'd like to believe it is useful nevertheless. Ceroklis 15:12, 29 September 2007 (UTC)

[edit] context setting

I added some desperately needed initial context-setting. One MUST immediately tell the lay reader who finds this page that mathematics is what it's about. Mathematicians would know that; others might not, unless it says so. Michael Hardy 23:10, 30 September 2007 (UTC)

[edit] Content error

The statement '26. A semi-modular lattice is atomic.[11]' in the article (and thus also the picture) is wrong. I do not have the given reference by hand, but a semi-modular lattice does not at all need to be atomic. However, a geometric lattice will always be semi-modular, atomic and relatively complemented. My reference is Richard Stanley's 'Enumerative combinatorics', page 104-105. —Preceding unsigned comment added by 130.237.48.107 (talk) 10:25, 8 October 2007 (UTC)

Statement 26 come straight from the reference in the article.
These are the definitions used there:
  1. Let L be a lattice. An element O\in{}L such that O\leq{}x\quad\forall{}x\in{}L is called the null element of L. It can be shown that the null element, if it exists, is unique.
  2. Let L be a lattice. If x > y but x > z > y is not satisfied for any z\in{}L we say that x covers y. We write x\succ{}y, or y\prec{}x.
  3. Let L be a lattice with a O. An atom is an element which covers O.
  4. An atomic lattice is a lattice with a O in which each element other than O contains at least one atom.
  5. A lattice L is semi-modular if it has no infinite chain and if \forall{}x,y\in{}L\quad{}x \prec x \vee y \Rightarrow x \wedge y \prec y.
I have looked in PlanetMath and have found the following definitions:
  1. idem
  2. idem
  3. Let L be a lattice. An atom is an element that covers some minimal element of L.
  4. A lattice L is atomic if any non-minimal element of L contains at least one atom.
  5. A lattice L is semi-modular if \forall{}x,y\in{}L\quad{}x \prec x \vee y \Rightarrow x \wedge y \prec y.
I'll need some time to figure out the consequences of the planetmath definitions but clearly these differences (most importantly the finite chain condition) have an impact. I presume I should move to these definitions and abandon my (somewhat outdated, I admit) reference. Ceroklis 15:23, 8 October 2007 (UTC)