Talk:Modular lattice

SVG images

I have added three SVG images to the article. I think the first should be unproblematic. With the other two I ran into a problem with broken Unicode support. I tried to include ∧ and ∨ (using the codes ∧ and ∨), but they only displayed as boxes. Therefore I had to draw these symbols. I hope that the standard font in SVG is sufficiently standardised that this is not going to be a problem. If you are seeing unusual spacing in the formulas in these pictures, please tell me about the problem so I can address it or rethink my approach, as appropriate.

Of course, if you have a solution for using Unicode in SVG and can tell me how to do it, that's even better. --Hans Adler (talk) 14:03, 10 February 2008 (UTC)

It's a problem, your symbols appear as boxes, even in the editable text. Hope you can fix it, I don't know how. SteveWoolf (talk) 19:15, 15 December 2008 (UTC)

They look ok to me. You must not have a full set of unicode fonts installed. —David Eppstein (talk) 19:19, 15 December 2008 (UTC)

undefined symbols

the lede posts a formula without defining the symbols involved first. I will probably come later to fix that but if someone wants to start already...  franklin  23:33, 10 February 2010 (UTC)

I don't think it makes much sense to define the operations here. A reader who doesn't know what a lattice is is not going to be illuminated by such an explanation. A reader who does know what a lattice is should be familiar with the notation. The lead has a prominent link to lattice (order), which of course defines the operations. Hans Adler 01:47, 11 February 2010 (UTC)

• It doesn't even say that those symbols are the operations of the lattice. It doesn't even read as a technical writing in math less for Wikipedia.  franklin  02:10, 11 February 2010 (UTC)

The symbol a remains undefined even now. Perhaps we need to add "for any element a" to the formula? yoyo (talk) 03:40, 21 January 2017 (UTC)

Why the name?

The article states that the submodules of a module form a modular lattice — is that where the word "modular" in "modular lattice" comes from? Is that the original basis for studying this class of lattices? If so we should say so but it would need a source. —David Eppstein (talk) 16:08, 20 March 2010 (UTC)

Yes and yes, but I forgot where I read that. I agree we should say so explicitly. Looking for a source now. Hans Adler 17:01, 20 March 2010 (UTC)

Actually I am no longer sure. I am now guessing that modular lattices got their name in the same way as (I believe) modules: From calculations modulo a number. When Dedekind first defined (modular) lattices – he called them "dual groups" –, he was working in the context of cosets of natural numbers modulo a fixed number. The title of his 1897 paper translates as "On decompositions of numbers by their greatest common divisors". Hans Adler 17:15, 20 March 2010 (UTC)

I have added a new reference (the book by Leo Corrie) that answers the question. Unfortunately the answer as presented in the book is a bit complicated. It looks to me as if Dedekind was interested in both aspects, and since he published some of his results with a great delay (after his retirement) we don't know about the precise order.

• He first described the modular law in his 1894 work on the theory of ideals, but didn't name it(?). There was a more complicated equation which he did name "modular identity".
• What we now call the modular identity first appeared in his 1897 paper, in which he examined the "Dualgruppe" ("dual group") formed by the divisors of a natural number with operations gcd and lcm, and in a digression defined dual groups (lattices) in general. He called it the "modular law" because it holds for modules. He called modular lattices "Dualgruppen vom Modultypus" ("dual groups of module type") because lattices of submodules have this property.

I will try to make a reasonable historical section out of this. Hans Adler 18:04, 20 March 2010 (UTC)

Finished. I am quite happy with the result. Of course it would be great if you could go over it to see if there are any obvious blunders. Hans Adler 18:43, 20 March 2010 (UTC)

One minor issue: according to Schlimm (doi:10.1007/s11229-009-9667-9) Dedekind first described a version of the modular law (in the context of modules rather than lattices) in 1877: Über die Anzahl der Ideal-Klassen in den verschiedenen Ordnungen eines endlichen Körpers. In Festschrift der Technischen Hochschule Braunschweig zur Säkularfeier des Geburtstages von C.F. Gauss (pp. 1–55). —David Eppstein (talk) 21:03, 20 March 2010 (UTC)

Thanks! I can't read the source from home, but in the meantime I have added a hint that Dedekind didn't publish all relevant papers after his retirement. Hans Adler 22:47, 20 March 2010 (UTC)

Hover link

The article gives the modular law as "a ≤ b implies a ∨ (x ∧ b) = (a ∨ x) ∧ b" which is fine. But if I hover over a link to Modular lattice, I see it as "a ≤ b implies a ∨ = ∧ b" which is nonsense. I wonder if there's a way to fix this? Maproom (talk) 09:42, 1 December 2021 (UTC)

This may be a problem with "=" appearing in the short description. I would fix it, but the short description seems to be imported from god knows where. - Jochen Burghardt (talk) 12:22, 1 December 2021 (UTC)

Inconsistent notation

Would it be too much to ask that the N₅ "Example" (and associated figure) use the same notation as the "Introduction". That is, a≤b implies forall.x ...

Same comment for the notation in the "Properties" section, where now we have a "c", instead of an "x", and the identification of {a,b,c} is an unnecessary permutation of the earlier introduced {a,b,x}.

Same comment for the labeling of the counterexample figure in the "Diamond isomorphism theorem" section, where we have x≤b, rather than a≤b.

These notational inconsistencies are a serious impediment to those who have come to the article to learn the concept. ScriboErgoSum (talk) 05:27, 19 March 2023 (UTC)

(Right) modular elements

The article currently defines a modular element to be a right modular element. While I am aware that certain sources make this definition, they do so in the context of geometric or semimodular lattices, where right modularity implies left modularity. Other sources (see e.g. Stanley Supersolvable lattices or Schmidt Subgroup lattices of groups) define a modular element to be one that is both right and left modular. Comment that since right modularity is not a self-dual property, there are two variants of any definition involving right modularity. Perhaps it would be better to mention the inconsistency in the article, or else to avoid the terminology altogether? Russ Woodroofe (talk) 18:06, 22 December 2023 (UTC)

Boldly implemented: changed "modular element" to "right modular element", and also defined left modular element. I've found also authors (e.g. Orlik and Terao) that define a modular element to be a left modular one. This portion of the article is mostly unsourced and has some other problems, which I did not try to fix just now. Russ Woodroofe (talk) 20:11, 28 December 2023 (UTC)

I appreciate your recent edit (I'm not an expert in the field of modular lattices, however). Could you add some text (footnotes) to mention the alternate terminology ("in the context of geometric or semimodular lattices" vs. Stanley, Schmidt from above; and maybe more variants you are aware of)? - Jochen Burghardt (talk) 09:45, 29 December 2023 (UTC)

Jochen Burghardt, I tried to follow your suggestion, while keeping with the general style of the rest of the article. Is this about what you were thinking? Separately, it might also be worthwhile (and a small change) to make left modularity a little more prominent in the article; my sense is that it is the more important of the two notions. Russ Woodroofe (talk) 17:28, 30 December 2023 (UTC)