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Is there a function $f:\Bbb N_+\to\cal P(\Bbb N_+)$ such that:

  • $f(a)\cap f(b)=f(\gcd(a,b))$,
  • $f(a)\cup f(b)=f(\operatorname{lcm}(a,b))$,
  • $a\in f(a)$, and
  • $f$ is injective?

Without the third condition, the function that maps a number to the set of its prime-power divisors works. Without the fourth, the trivial map $x\mapsto\Bbb N$ works.

From either the first or the second condition, we get $f(a)\subseteq f(ab)$. Combining with the third, we get that $f(a)$ contains the set of $a$'s divisors.

I was also able to show that $6\in f(2)$ iff $6\in f(10)$. But I really have no idea how to go from here, or even if such a function exists.

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  • $\begingroup$ If you remove the second condition then the set of divisors works, if you remove the first, then the function $f(x)=\mathbb N \setminus\{2k : x|k\}$ works. $\endgroup$
    – Asinomás
    Jun 3, 2016 at 16:39
  • $\begingroup$ @arctictern Fixed, thanks $\endgroup$
    – Akiva Weinberger
    Jun 3, 2016 at 20:04
  • $\begingroup$ When $\Bbb N$ is partially ordered by the divisibility relation, one may describe $f$ as a lattice embedding $\Bbb N\to P(\Bbb N)$ with the extra property $x\in f(x)$. (Just for some terminology.) $\endgroup$
    – anon
    Jun 3, 2016 at 20:07

1 Answer 1

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Let $f(1)$ be the set of all naturals which are not prime powers. Then define $f(n)$ to be $f(1)$ in union with the set of all prime power divisors of $n$.

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