I just wonder what is the non-planar posets (reps.lattices) with least amount of elements? Or where can I find such results? Thanks a lot...
By a non-planar poset I mean it admits no Hasse diagram in which no two edges intersect. A sufficient condition for the Hasse diagram to be non-planar is that it is non-planar as a graph, but this is not necessary.
If by non-planar, you mean with a non-planar Hasse diagram, then using Kuratowski's Theorem, it must contain a subgraph that is a subdivision of $K_5$ or of $K_{3,3}$. So it must have at least 5 vertices : given that $K_5$ cannot represent a poset (it contradicts transitivity), then the smallest one is $K_{3,3}$ and achieved by the following poset, ordered by inclusion :
$$ \large\{ \{1\},\{2\},\{3\},\{1,2,3,4\},\{1,2,3,5\},\{1,2,3,6\} \large\} $$
This poset is not a lattice though. I would expect the smallest one to be $$ \large\{ \emptyset, \{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\},\{1,2,3\} \large\} $$