Find lattice corresponding to sublattice

Question

I have a question concerning Lattice and sublattice.

I need to find the lattice corresponding to a lattice of sublattice. I have pasted the whole exercice. (It is the exercice 2.12 of the book Davey & Priestley). It's the question (ii):

I guess that the number of atoms of the lattice is 4. But I have not been able to find the lattice corresponding to $Sub_0(L)$.

Thank you.

Answer 1

Hint. The bottom element corresponds to $\varnothing$.
Since the lattice operations are idempotent, each element alone is a sublattice;
so if $\mathrm{Sub}_0 L$ has $4$ atoms, it means that $L$ has $4$ elements.
Now, how many (up to isomorphism) lattices with four elements are there?
It's only a matter of trying those few, and only one works.

Answer 2

This is L:

Right? 3 et 2 are comparable (by transitivity) but not 1 and 4.