I have the following exercise:
Having the lattice $D_{165}$ of the divisors of 165, ordered by divisibility
1) Draw Hasse Diagram;
2) Find all complements;
3) Check if it's a distributive lattice;
4) Check if it's a Boolean lattice.
My development:
The divisors of 165 are $D_{165} = \left \{ 1,3,5,11,15,33,55,165 \right \}$
1)
this is the diagram I have done, look at the figure on the left:
2)
in this table I have put an asterisk when an element has its complement.
i.e. every time is true this condition $\forall a \in D_{165}$
$a \land a' = 0 \quad \mbox{ and } \quad a \lor a' = 1$
\begin{array}{c|c}
\, & 1 & 3 & 5 & 11 & 15 & 33 & 55 & 165 \\
\hline 1 & - & - & - & - & - & - & - & * \\
\hline 3 & - & - & - & - & - & - & * & - \\
\hline 5 & - & - & - & - & - & * & - & - \\
\hline 11 & - & - & - & - & * & - & - & - \\
\hline 15 & - & - & - & * & - & - & - & - \\
\hline 33 & - & - & * & - & - & - & - & - \\
\hline 55 & - & * & - & - & - & - & - & - \\
\hline 165 & * & - & - & - & - & - & - & - \\
\end{array}
so this lattice is NOT a Complemented lattice.
3)
it is NOT possible to get a sublattice like one of the two nondistributive lattices on the right in the previous image.
So the lattice $D_{165}$ is distributive.
4)
Since NOT all elements in the lattice have a complement, i.e. it is not a Complemented lattice. The lattice $D_{165}$ is not a Boolean lattice.
Please, can you tell me if the exercise is correct? Can you give me any suggestion?
Many thanks!
By re-arranging the elements of the Hasse diagram, you can check that it's isomorphic to the 8 elements Boolean lattice (aka, the cube).
So it is indeed complemented, and I suppose it answers all of your queries.
By the way, in the same page I put a link above, in the section examples, you can see that whenever $n$ is a square-free number, as it is the case of $165$, the lattice of divisors is Boolean.
Clearly this is a necessary and sufficient condition.