Suppose we have a a bounded lattice $L$. We might ask: does there exist a non-trivial linear representation of $L$, i.e. a lattice homomorphism $\rho: L \to \text{Sp}(V)$, where $V$ is a non-zero finite dimensional vector space, and $\text{Sp}(V)$ is the lattice of subspaces of $V$, and we require $\rho(0) = \{ 0\}$ and $\rho(1) = V$.

One simple necessary condition is the existence of a function $d: L \to \mathbb{N}$, satisfying:

- $d(0) = 0$, and $d(1) = n > 0$
- $d(a) \leq d(b)$ whenever $a \leq b$
- $d(a) + d(b) = d(a\lor b) + d(a\land b)$

Is the existence of such a $d$ sufficient?

I've encountered various related results about *faithful* representations (i.e. sublattices of $\text{Sp}(V)$). E.g. those lattices need to be modular and arguesian. However, for representations that are not required to be faithful, we might get away with non-modularity for example, by just quotienting out the non-modular part of the lattice (e.g. collapsing the vertical edge in the non-modular pentagon $N_5$). I haven't found any literature on what sort of conditions might remain if we don't require faithfulness. Are there any known results regarding this?