I recently found out that there is a link between planar graphs and distributive lattices, e.g. certain sets of orientations of a planar graph form distributive lattices. When i read a theorem stating that a lattice is distributive if and only if it has no sublattice isomorphic to N5 (the pentagon lattice) and M3 (the diamond lattice) it struck me that there is a huge apparent similarity to the planarity argument in graphs where a graph is planar (on S0) if and only if it doesn't not contain a subdivision of the graphs K5 (the complete graph with 5 vertices) and B3,3 (the complete bipartite graph with 2 partitions with 3 vertices each).

Since my actual work is in a different field i cannot spare the time to really investigate this on my own, yet i am very interested in a possible connection here. So does anyone here know of a paper/thesis or similar that expounds on or disproves this apparent similarity?