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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.

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    $\begingroup$ What is the definition of quasi-sublattice? $\endgroup$ – Richard Stanley Nov 8 '19 at 2:16
  • $\begingroup$ @RichardStanley a quasi-sublattice is a subset closed under meets and joins but not necessarily empty meets and joins (i.e the top and bottom element of the lattice) $\endgroup$ – Hans Nov 8 '19 at 20:33
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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 $\mathrm{Gr}(V)$.

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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

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  • $\begingroup$ thanks for sharing this result. do you recall the definitions of a proper line or a proper projective plane? $\endgroup$ – Hans Jun 10 '20 at 17:55
  • $\begingroup$ 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? $\endgroup$ – Hans Jun 10 '20 at 17:57
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    $\begingroup$ @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. $\endgroup$ – Sam Hopkins Jun 11 '20 at 11:24
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    $\begingroup$ 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". $\endgroup$ – Jukka Kohonen Jan 26 at 17:28
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    $\begingroup$ The answer is fine, just noting that Cameron's "atomic" is the same as Stanley's. $\endgroup$ – Jukka Kohonen Jan 26 at 17:36

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