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Birkhoff's representation theorem implies that every distributive lattice embeds into the lattice of subsets of a set. Is there also some representation theorem for modular lattices?

For example, I wonder: Does every modular lattice embed into the lattice of submodules of some module? After all, I cannot think of any relation which holds in the lattice of submodules of some module, except for modularity.

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  • $\begingroup$ I have accepted Todd's answer, but if there is any more sophisticated representation theorem for modular lattices (different from the naive one I have suggested), feel free to add an answer! $\endgroup$
    – Martin Brandenburg
    Commented Feb 8, 2016 at 17:07
  • $\begingroup$ Perhaps a more direct analogue of Birkhoff's representation theorem is the representation of geometric lattices by matroids: en.wikipedia.org/wiki/Geometric_lattice $\endgroup$
    – François G. Dorais
    Commented Feb 16, 2016 at 5:36
  • $\begingroup$ There's also von Neumann's coordinatization theorem, which is close to what the OP suggests. Specifically, every complemented modular lattice of order at least 4 is isomorphic to the lattice of principal right ideals of a von Neumann regular ring - see en.m.wikipedia.org/wiki/Continuous_geometry $\endgroup$
    – Tristan Bice
    Commented Feb 19, 2016 at 2:41
  • $\begingroup$ @TristanBice: What about making this an answer? :) $\endgroup$
    – Martin Brandenburg
    Commented Feb 19, 2016 at 7:59
  • $\begingroup$ Birkhoff’s theorem, strictly speaking, is about finite distributive lattices. The fact that finite modular lattices which satisfy one additional property (can be: complemented or atomic or $1$ is the join of atoms) are projective geometries seems similar to me. I think this is also due to Birkhoff (as I asked about in another MO question). $\endgroup$
    – Sam Hopkins
    Commented Oct 16 at 20:44

4 Answers 4

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There are lots of relations satisfied in lattices of submodules besides the ones implied by modularity. For example, there is the Desarguesian identity mentioned here (which holds in any lattice of congruence relations on an algebra of a Mal'cev theory, such as the theory of modules over a ring).

So, a projective plane which is not Desarguesian gives an example of a modular lattice which is not faithfully represented in a lattice of submodules (to the collection of points and lines, adjoin a formal top and bottom to obtain a lattice). See remarks by Freyd-Scedrov in Categories, Allegories 2.156-2.157, here.

Similarly, the free modular lattice on four generators admits no faithful representation into a lattice of submodules. If you have access to the page, see Theorem 3 on page 145 of this Google book.

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  • $\begingroup$ Saw the answer notification just when hitting Enter! $\endgroup$
    – Pedro Sánchez Terraf
    Commented Feb 8, 2016 at 13:17
  • $\begingroup$ @Todd You wrote that there are lots of other relations, and the Arguesian law is just one example. What are the other relations? $\endgroup$
    – Martin Brandenburg
    Commented Oct 18 at 12:08
  • $\begingroup$ @MartinBrandenburg This is a hard question to answer, and I am no expert. However, each submodule defines a congruence (an internal equivalence relation in the category of modules) by taking cosets, and these equivalence relations commute under relational composition. Therefore we are in the presence of a linear lattice, in the sense of Rota. The thesis of Marc Haiman, scholar.google.com/scholar?cluster=12728508576024553972, gives a graphical procedure for generating identities that hold in linear lattices. The first of the higher Arguesian identities appears on page 73. $\endgroup$
    – Todd Trimble
    Commented Oct 19 at 1:22
  • $\begingroup$ I believe this is really closely related to passages in Freyd-Scedrov which start at 2.156 and continue past 2.157; unfortunately I cannot find my copy of the book. But my memory is that they present a similar sort of graphical calculus for identities in one-object allegories in which composition of morphisms gives their lattice theoretic join, and this is close to the same thing. Someone really ought to look carefully into these connections. There is probably a PhD thesis in there. (All this is also closely connected to Mal'cev theories.) $\endgroup$
    – Todd Trimble
    Commented Oct 19 at 1:28
  • $\begingroup$ Finally, this survey arxiv.org/pdf/2403.19677 seems to give useful pointers to the theory of linear lattices. $\endgroup$
    – Todd Trimble
    Commented Oct 19 at 2:16
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The lattice of submodules of a module satisfies a stronger identity, namely the Arguesian law: $$ (x_0\vee y_0)\wedge (x_1\vee y_1)\wedge (x_2\vee y_2)\leq ((z\vee x_1)\wedge x_0) \vee ((z\vee y_1)\wedge y_0), $$ where $ z := z_2 \wedge (z_0 \vee z_1)$ and $z_i := (x_j \vee x_k) \wedge (y_j \vee y_k)$ for $\{i,j,k\} = \{0,1,2\}$.

You might check the short paper

Day, Alan; Jónsson, Bjarni. The structure of non-Arguesian lattices. Bull. Amer. Math. Soc. (N.S.) 13 (1985), no. 2, 157--159.

where a “geometric” characterization of modular, non-arguesian lattices is proved.

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  • $\begingroup$ What are these strange symbols in the formula? They are not shown properly at my computer. $\endgroup$
    – Martin Brandenburg
    Commented Feb 8, 2016 at 17:05
  • $\begingroup$ @MartinBrandenburg Strange, here the formula looks fine but the editor seems to have inserted some invisible characters (I found that by editing the answer and C&P into a text editor. In additive notation, it is $(x_0+ y_0)\cdot (x_1+ y_1)\cdot (x_2+ y_2)\leq ((z+ x_1)\cdot x_0) + ((z+ y_1)\cdot y_0).$ I'll try to erase the strange chars now. $\endgroup$
    – Pedro Sánchez Terraf
    Commented Feb 8, 2016 at 18:44
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    $\begingroup$ @MartinBrandenburg I've just edited. Please tell me if it looks better. If it does, this might be a bug on the site. $\endgroup$
    – Pedro Sánchez Terraf
    Commented Feb 8, 2016 at 18:49
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Faigle and Herrmann, "Projective Geometry on Partially Ordered Sets," Transactions of the American Mathematical Society 266 (1981), 319-332. (There is an error in the statement of Corollary 4.8.) http://www.ams.org/journals/tran/1981-266-01/S0002-9947-1981-0613799-9/S0002-9947-1981-0613799-9.pdf

There are also articles, a book and a thesis by Stefan E. Schmidt on topics like "projective geometry on an ordered set of points."

Edit: Schmidt, Stefan E. Grundlegungen zu einer allgemeinen affinen Geometrie. Birkhäuser Verlag, Basel, 1995.

Schmidt, Stefan E. Projektive Räume mit geordneter Punktmenge. Mitteilungen aus dem Mathematischen Seminar Giessen, No. 182 (1987).

There is also Benson, D. J. and Conway, J. H., "Diagrams for modular lattices," J. Pure Appl. Algebra 37 (1985), no. 2, 111–116, although its representation is not as good as Faigle and Herrmann's. I just list it since it's Conway.

I haven't read J. Yves Semegni's thesis, "ON THE COMPUTATION OF FREELY GENERATED MODULAR LATTICES," but it discusses a representation in Section 5.4. https://scholar.sun.ac.za/handle/10019.1/1207

Then there is Marcel Wild's unpublished manuscript, "Modular Lattices of Finite Length." It is available on his website.

Further edit: These references give representation theorems for modular lattices, at least those of finite height. They do not give a representation theorem in terms of lattices of submodules, for the reasons others have stated, but they do give representations in terms of something like a projective geometry on a partially ordered set of points.

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    $\begingroup$ It would help me and others in how far these references answer my question, and what are the main results relevant for my question. $\endgroup$
    – Martin Brandenburg
    Commented Feb 16, 2016 at 7:49
  • $\begingroup$ I upvoted this answer years ago, but I came across it again and I just wanted to comment on how much I learned from it. $\endgroup$
    – arsmath
    Commented Aug 24, 2018 at 21:03
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    $\begingroup$ Arsmath, my work here is done. (Flies off, cape swooshing in the wind.) $\endgroup$
    – Tri
    Commented Aug 25, 2018 at 14:28
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By adding a couple of conditions we can indeed obtain a representation theorem as the OP suggests. Specifically, von Neumann's coordinatization theorem says that every complemented modular lattice of order at least 4 is isomorphic to the lattice of principal right ideals of a von Neumann regular ring. See the wikipedia article on continuous geometry or any of the books on continuous geometry mentioned in this mathoverflow post.

If we replace modularity with the stronger Arguesian law mentioned in Todd and Pedro's answers, we get the more general coordinatization theorem of Jónsson, which says that every complemented Arguesian lattice with a large partial 3-frame is again isomorphic to the lattice of principal right ideals of a von Neumann regular ring. See Proposition 10.1 of this paper by Wehrung for a first order characterization of "having a large partial 3-frame".

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