Function that turns GCD and LCM into intersections and unions?: $f(a)\cap f(b)=f(\gcd(a,b))$, $f(a)\cup f(b)=f(\operatorname{lcm}(a,b))$

Question

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.

Answer 1

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 $n$.