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There are various "representation theorems" for lattices such as Birkhoff's Representation Theorem that states that every finite distributive lattice is isomorphic to a quasi-sublattice of the lattice of subsets of some set. There is an analogous Stone Representation Theorem for Boolean algebras.

Here is a fact that inspires this question: given a vector space $V$, the Grassmanian, $\mathrm{Gr}(V)$, is the lattice of subspaces of $V$ where joins are given by subspace sum, and meets by intersection. Fact: $\mathrm{Gr}(V)$ is a modular lattice (this holds more generally for the lattice of subgroups of an abelian group, for example).

Here is my first question: given a modular lattice, $L$, does there exist a vector space $V$ such that $L$ is isomorpic to a quasi-sublattice of $\mathrm{Gr}(V)$? This post may be a partial answer to this question, but perhaps there is more progress towards an answer in less generality.

Hans
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2 Answers2

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The first question needs some kind of drastic modification to have a chance of being true. For instance, let $L$ (respectively, $M$) be the lattice of subspaces of an $n$-dimensional vector space over $\mathbb{F}_2$ (respectively, $\mathbb{F}_3$), where $n\geq 3$. Then the ordinal sum of $L$ and $M$ (just "stack" $M$ on top of $L$) is not a quasi-sublattice of any $\mathrm{Gr}(V)$.

Richard Stanley
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Maybe you are aware of this, but there is a representation theorem for atomic modular lattices. Below I captured a picture of Theorem 7.56 on pg. 288 of Peter Cameron's "Introduction to Algebra":

enter image description here

Sam Hopkins
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  • thanks for sharing this result. do you recall the definitions of a proper line or a proper projective plane? – Hans Jun 10 '20 at 17:55
  • addendum: in particular, since every finite lattice is trivially atomic, every finite modular lattice is classified as such... I also wonder if there is any particular structure to the direct product decomposition? – Hans Jun 10 '20 at 17:57
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    @Hans: line/projective plane have their usual meaning (a line is lattice which has a maximal element, a minimal element, and some number of atoms; a projective plane is a rank 3 lattice following the usual rules of lines=coatoms/points=atoms). But it is far from true that every finite lattice is atomic: here atomic means generated by atoms (this is also sometimes called 'atomistic,' I guess). E.g., the 5 element "pentagon" lattice is not atomic. – Sam Hopkins Jun 11 '20 at 11:24
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    Unfortunately "atomic" has two meanings in lattices. For some (e.g. Stanley in Enumerative Combinatorics) "atomic" means every element is a join of atoms. For some (e.g. Grätzer in Lattice Theory: Foundation) "atomic" means every element majorizes an atom (trivially true in all finite lattices), and the previous thing is "atomistic". – Jukka Kohonen Jan 26 '21 at 17:28
  • @JukkaKohonen: that's true. Does it affect the answer, though? – Sam Hopkins Jan 26 '21 at 17:30
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    The answer is fine, just noting that Cameron's "atomic" is the same as Stanley's. – Jukka Kohonen Jan 26 '21 at 17:36