I'm trying to determine all the prime filters in the lattice $N = (\mathbb{N}, \text{lcm}, \text{gcd})$, where the order is given by divisibility.
I know that this lattice is distributive and that its join-irreducible elements are precisely the positive powers of any prime number. I have proved that for any join-irreducible element $x \in N$ it is the case that $N \backslash \uparrow x$ is a prime ideal. Since $\uparrow x$ is a prime filter, I'm wondering if every prime filter in $N$ has this form. Any ideas?