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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?

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$\newcommand{\lcm}{\operatorname{lcm}}$Suppose that $F$ is a (proper) prime filter in $N$. Let $m\in F$ be minimal. If $m$ is not a prime power, it has a non-trivial factorization $m=ab$ such that $\gcd(a,b)=1$. Then $m=\lcm(a,b)$, so $a\in F$ or $b\in F$, contradicting the minimality of $m$. Thus, every minimal element of $F$ is a prime power, and it follows immediately that $F={\uparrow m}$.

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That's a cute argument, Brian. Thanks! – ragrigg Nov 7 '12 at 6:41
@ragrigg: You’re welcome! – Brian M. Scott Nov 7 '12 at 8:00

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