I have been struggling for several days on this question, now it is time for you folks to enlighten me :)
In the book Lattice Theory: Foundations from George Gratzer (pdf file), it is stated, page 9, the following :
Define the order-dimension, dim(P), of an order P as the smallest cardinal m such that P is a suborder of a product of m chains. For a finite lattice L, planarity is equivalent to dim(L) ≤ 2
From what I understand : if I take any planar Hasse diagram, I should be able to represent it as a subdiagram of a diagram representing the product of at most 2 chains.
Let us consider the following planar diagram :
It is clearly not a subdiagram of a Hasse diagram representing one chain... So it should be a subdiagram of a Hasse diagram representing the product of two chains (as it is a planar diagram).
A product of two chains basically looks like that (here, it represents the product of a chain of 4 elements over a chain of 3 elements) :
Which brings me to my question : I do not picture any diagram that would represent a product of two chains, and whose any subdiagram would look like the planar one I introduced first. For example, in a diagram representing a product of two chains, there is no such a vertex gathering 4 edges going down as it is the case for the top vertex in the former planar diagram. This is in contradiction with the Gratzer statement I copied at the beginning of this post.
If anybody can explain where I am wrong, I would really be thankful !