Talk:Map of lattices

Announcement, antimatroids, etc.

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)

I've added some more material to the antimatroid article that makes it clear that they are the same thing as join-distributive lattices (in the same way that in your example power sets are the same thing as complete boolean algebras). In that framework they are defined in purely lattice-theoretic terms and would fit under distributive lattices and above semimodular lattices. —David Eppstein (talk) 01:13, 1 February 2009 (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.

Ceroklis: Why do you make such claims? Two students, Stefan Reeg and Wolfgang Weiß, did the full exploration for over 50 properties, with all necessary proofs and examples. Their work is unpublished, and it does not cover what is needed here, because they consider only finite lattices. But their counterexamples can certainly be used. I do not expect "an enormous amount of work" here. --Bernhard Ganter (talk) 14:25, 17 December 2017 (UTC)

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)

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)

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)

I've seen a different definition, in which a lattice is atomic iff every element is the join of the atoms it contains. According to arxiv:0712.1047 (which uses that definition) a semimodular lattice need not be atomic, and a geometric lattice is one that is both semimodular and atomic. However that definition seems to be more frequently associated with the different word "atomistic". —David Eppstein (talk) 23:01, 1 February 2009 (UTC)

The diagram should be removed

As said above the diagram is with an error. It should be removed from Map of lattices page as well as List of algebraic structures page until it will be corrected.

Erroneous diagram is worse than no diagram. —Preceding unsigned comment added by Porton (talk • contribs) 12:02, 20 June 2009 (UTC)

Just because it says there is an error doesn't mean there is one. In fact, the diagram is perfectly correct and consistent with the definitions of the reference given in the article. And this set of definitions is no worse nor better than any other.

The problem is that different authors define the different types of lattices differently. What should be done is to open a discussion in the context of the math project in order to decide on one set of definitions for lattices. It might be those of a specific book, those of planetmath, whatever. Then, and only then, would it make sense to correct all articles on lattice theory (including this one) in order to conform to the chosen definitions. Reaching a reasonable consensus is a significant effort. If you want to launch it, go ahead. But until this is done there is no point in touching this particular page (which is at least self-consistent) since most pages on lattice theory do not adhere to a common convention. Ceroklis (talk) 03:01, 21 June 2009 (UTC)

Not every Boolean/Heyting Algebra is atomic. For instance, the Lindenbaum–Tarski algebra for propositional logic is a Boolean algebra but is not atomic. Likewise, the Lindenbaum Algebra for intuitionistic logic is a non-atomic Heyting Algebra. — Preceding unsigned comment added by 98.220.235.232 (talk) 22:24, 29 April 2012 (UTC)

A total order is always Heyting, not just distributive. 213.112.229.130 (talk) 08:21, 21 June 2017 (UTC)

(May 2017) {{Disputed}} tags

This claim has been under dispute for almost ten years(!) without being resolved or even acknowledged in the article. I've remedied the latter by adding {{Disputed}} tags to the article. Robin S (talk) 18:39, 27 May 2017 (UTC)

Great article

I'm a student, and I can testify this diagram and the article, and the inclusion proofs below, are an amazingly useful resource for someone trying to learn, study and organize themselves, thanks for doing this, please feel appreciated because your work IS VALUABLE! Please don't delete this article, it's great. — Preceding unsigned comment added by Santropedro (talk • contribs) 17:30, 24 March 2017 (UTC)

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