Divisibility can be cast in terms of subsets of the prime factor multisets. That is, for every number, we can translate it to its multiset of prime factors, e.g

Divisibility can be cast in terms of subsets of the prime factor multisets. That is, for every number, we can translate it to its multiset of prime factors, e.g. 36 would become {2->2, 3->2}; then divisibility becomes the subset relation. In this world, 1 is the empty set, primes are the singleton sets, and composites are anything with at least two elements. 0, being divisible by everything, is a superset of everything; it probably makes the most sense to treat it as {p->aleph_0} for all p, though it's rather undetermined -- you could pick another cardinal if you wanted. The inclusion of negative numbers would basically freely adjoin a copy of this lattice with itself. You'd get two "empty-ish" things that are divisible by each other, two ranks of "singletons" that are divisible across the signs, and so on. Modding out by these cycles gets you the original lattice. There may be some value to keeping the negative numbers around -- certainly in category theory we like to keep equivalences explicit rather than modding out by them -- but "morally" you don't get any new relations by adding the negative numbers.