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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.

My second question concerns cellular sheaves: let $\mathcal{F}: X \rightarrow \mathbf{Mlc}$ be a functor from the face relation poset of a cell complex (i.e. $\sigma \trianglelefteq \tau$ if $\sigma$ is a face of $\tau$ with transitive closure...) to the category, $\mathbf{Mlc}$, of modular lattices and (Galois) connections. Does there exist a cellular sheaf $F: X \rightarrow \mathbf{Vec}$ such that $\mathcal{F} \cong \mathrm{Gr} \circ F$? In a way, this question is the natural generalization to the first question to a particular class of simple diagrams of modular lattices and connections.

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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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