Take $K$ to be a field and take $L$ to be a finite meet-semilattice. I'm interested in the set of functions $n: L \rightarrow \mathbb{Z}^{\ge 0}$ such that there is some function $V$ from $L$ to vector spaces over $K$ so that
\begin{align*} \dim V(a) = n(a) \qquad\qquad&\text{ for}\quad a \in L \,\,\,\text{ and}\\ V(a \wedge b) = V(a) \cap V(b) \quad& \text{ for} \quad a, b \in L. \end{align*}
In particular, I'm interested in the following:
There are some necessary conditions on $n$; it needs to satisfy $n(a) \ge n(b)$ for $a \ge b$, and it needs to be supermodular: $$n(a) + n(b) \le n(a \vee b) + n(a \wedge b)\quad\text{for}\quad a,b \in L.$$ For which $L$ and $K$ are these the only conditions we need? In particular, can it be done for $L$ the semilattice associated of a modular arguesian lattice and $K$ algebraically closed?
Taking direct sums shows that, if $n_1$ and $n_2$ obey the conditions given above, then so does $n_1 + n_2$, so that the set of $n$ form a monoid. If $K$ is a finite field, is this monoid finitely generated?
I'm also interested in the version of this question phrased for $L$ a lattice, where we also require the function $V$ to satisfy $$V(a \vee b) = V(a) + V(b).$$ In this case, any satisfying $n$ must be modular: $$n(a) + n(b) = n(a \vee b) + n(a \wedge b)\quad\text{for}\quad a,b \in L.$$ This seems to relate the problem to Wehrung's definition of a dimension monoid, but there does not seem to be research on when an element of the dimension monoid can be realized by a lattice of vector spaces.
My motivation is actually number theoretic: the characteristic $2$ case of these questions gives some information about how $2$-Selmer groups change in isogeny families.
