What does distributive property translate into in an non directed graphs?

Question

The distributive property of distributive lattice according my intuition should bring in some restriction so as to the kind of links we can have in our hasse diagram. Is it so? Earlier I thought that distributive property might not allow for multiple paths from one node to another, but that obviously came out to be false. Can you explain what property does distributive law imply on hasse diagram (non directed graph).

Answer 1

Duffus and Rival characterized finite graphs that are Hasse diagrams of distributive lattices in their 1983 paper:

Theorem A finite graph $G$ is the covering graph of a distributive lattice of length $n$ if and only if $G$ is a retract of the hypercube $2^n$ and $\mathrm{diam}(G)=n$.

Here, $2^n$ is simply the Hasse diagram of the Boolean algebra with $n$ atoms.