Let $L$ be a lattice (we can assume that it is distributive), according to Birkhoff (page 72):
Two intervals of a lattice are called trasposes when they can be written as $[a \wedge b, a]$ and $[b, a \vee b]$ for suitable $a$ and $b$.
Likewise, two intervals $[x, y]$ and $[x', y']$ are called projective (in symbols $[x, y] \sim [x', y']$) if and only if there exists a finite sequence $[x, y]$, $[x_1, y_1]$, $[x_2, y_2]$, $\ldots$, $[x', y']$ in which any two succesive quotients are trasposes.
An interval $[a, b] = \{x \in L: a \leq x \leq b \}$ is called prime when $b$ covers $a$.
It could be seen that the binary relation of projectivity over the set of prime intervals is an equivalence relation.
My question is about two examples of distributive lattices.
In the first example, I want to see which are the equivalence classes of the lattice whose Hasse diagram is presented in this picture (where the nodes are labeled with letters; and edges are labeled with $e_i$, $1 \leq i \leq 42$):
It should be said that the projectivity relation it is not on the nodes of the Hasse diagram, but on the prime intervals of the Hasse diagram. So, it has not sense to write: ∼; it should be written expresions of the type [,]∼[d,e], because the projectivity relation ∼ is defined over prime intervals. The transitivity property should be understood on intervals, for example: if [,]∼[d,e] and [d,e]∼[g,h], then [,]∼[g,h].
Are [b, d] and [e, h] projective?
I think that they are not projective. Could someone confirm that?
If this is true, and we choose a color for each equivalence class of projective intervals: Is the following diagram correct?
The second example is the following:
Edit: I've changed the images of the second example
The questions are the same as in the first example. Is the following diagram correct?
Edit: I've heard somewhere that the number of equivalence classes of the projectivity relation, in a distributive lattice, equals the length of one maximal chain. Could someone give me a reference of this stament?
Thanks in advance.
Regarding your question about if [b, d] and [e, h] are projective, you are right, they are not projective.
Therefore, The picture of the fisrt example with colors, is correct.
The same can be said for the second example, the picture of the second example with colors is correct.
The theoretical justification, can be seen in the book that you have mentioned (Birkhoff, Lattice Theory) on page 76, exercise 1, where is stated that opposite sides of a quadrilater represent transposed prime intervals in a finite modular lattice. As your two examples are even distributive, the statement is true.
Regarding the reference you are requesting at the end, in the article "On the projective structure of a modular lattice" of Thrall, in the introduction is stated that in a distributive lattice, the number of equivalence classes under the relation of projectivity equals the length of any maximal chain.
Hope it will be useful!