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Dedekind proved that the free modular lattice on 3 generators is realisable by the intersections and sums of 4-dimensional subspaces in 8-space. Birkhoff showed that the free lattice is infinite if it has at least 4 generators. My questions:

  1. Is every free modular lattice realisable by subspaces in a vector space?

  2. If not, is there a description of the freest lattice consisting of subspaces of a vector space?

  3. Is there a nice description of symmetric elements in the free lattice? Namely, there is a natural action of Sym(n) on the n-generated free lattice; what are its fixed points? Are there more than (sums of k-fold intersections) for k=1...n?

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The answer to the first question is no. Bjarni Jónsson proved that lattices of modules must satisfy the Arguesian identity. This is the lattice theoretic analogue of Desargues' theorem in projective geometry. You can find a discussion of the result in this survey article about Jónsson's career.

If you are willing to consider complemented modular lattices, then the Arguesian identity exactly characterizes representability by vector spaces. This is essentially Frink's characterization of projective spaces.

There is another approach by I am less familiar with this approach, but I think another way you can see this is through some work of Gelfand and Ponomorov on the 4-subspace problem. They proved that there are elements of the free modular lattice that must equal either 1 or 0 on an indecomposable representation. You can see discussion of this approach in a recent paper by Stekolshchik.

For non-complemented lattices, I think the answer to the second question is unknown. There is a partial result by Mark Haiman that handles an easier problem of when a lattice is "type 1". A lattice is type 1 when it can be represented as a lattice of commuting equivalence relations. He gives an algorithmic characterization.

Haiman called these lattices "linear lattices", but Palfy and Szabo gave an example of a type 1 lattice that can be represented as a lattice of normal subgroups of a group, but cannot be represented as a lattice of subgroups of an abelian group. This covered in this survey article by Palfy.

I don't know the answer to the third question, though the free modular lattice on four generators has an undecidable word problem, so it might be intractable.

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  • $\begingroup$ Thanks! I should have been clearer in the third question that Iam interested in symmetric elements of free arguesian lattices, too. $\endgroup$
    – grok
    Commented Sep 5, 2018 at 19:25
  • $\begingroup$ The arguesian identity involves 7 variables, so the question remains about whether free modular lattices on $n\le 6$ generators are arguesian (obviously yes when $n\le 2$), maybe you know the answer? $\endgroup$
    – YCor
    Commented Sep 21, 2018 at 8:31
  • $\begingroup$ (Yes for $n=3$ by the construction at the end of golem.ph.utexas.edu/category/2015/09/…) $\endgroup$
    – YCor
    Commented Sep 21, 2018 at 8:56
  • $\begingroup$ Yes, the free modular lattice on 3 variables can be represented as a lattice of vector spaces (I think this goes all the way back to Dedekind). I think for 4 the answer is no, though I'm not 100% sure. Are non-Desarguesian projective planes 4-generated as lattices? That would show it. $\endgroup$
    – arsmath
    Commented Sep 21, 2018 at 12:21
  • $\begingroup$ IIUC, the arguesian identity involves 6 variables (and cannot be expressed with 5 variables) Haiman (link.springer.com/article/10.1007/BF01188053). Palfy can now be obtained at web.archive.org/web/20121022011544/https://math.hawaii.edu/… $\endgroup$
    – Dale
    Commented Dec 2, 2023 at 16:57

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