EDT: Clarification: How could one determine if $N$, according to the question in the title, is a lattice?
I have made the following assumption, which I wonder whether it is correct or not:
The set $N$ can be defined as $N:=\left(\mathbf{Z}_+, \mid\right)$ where $\mathbf{Z}_+ = \left\{1,\,2,\,3 \, ... \right\}$.
The number $1$ divides every number, and any prime number (according to definition) can only be divided by $1$ or itself.
Thus, if we draw a fragment of the Hasse diagram (just focusing on any of the two primes $p_i$ and $p_j$) that is corresponding to the set $N$, it could look something like:
Considering Euclid's proof of infinite primes - also considering that in our case, there are no limitations of how great of a value a prime number can have - we can assume that e.g. $p_i \rightarrow \infty$.
This leads us to the conclusion that:
$1 \cup p_i=\infty$
$p_i\cup p_j=p_ip_j=\infty$ (since $\forall \, k$ $p_k\geq 1$ and $p_i \rightarrow \infty$)
$\vdots$
Since, for every pair of elements which contains $p_i$ as defined above, there exist no least upper bound (here is where I am very unsure though) and the set $N$ therefore is not fulfilling the definition of a lattice - which is that there for every pair of elements must exist a least upper bound as well as a greatest lower bound. Therefore, $N$ can't be a lattice.
Am I on right track?