# Representations of modular lattices, extension to cellular sheaves

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.

• What is the definition of quasi-sublattice? – Richard Stanley Nov 8 '19 at 2:16
• @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) – Hans Nov 8 '19 at 20:33

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 $$\mathrm{Gr}(V)$$.