Classification of strongly lcm-closed sets

Question

I call a set X of positive integers strongly lcm-closed if a,b ∈ X if and only if lcm(a,b) ∈ X. In the finite case X is the set of divisors of lcm_{x ∈ X}x. But in the infinite case it is more interesting, for example, $\{a \geq 1: a \not\equiv 0 \pmod p\}$ and $\{p^a:a \geq 0\}$ for any prime p, are strongly lcm-closed sets.

Which sets are strongly lcm-closed sets?

This question arose in my Ph.D. thesis (p.107) where strongly lcm-closed sets describe where autotopisms of Latin squares give rise to subsquares.

As a side question:

Is there a common name for strongly lcm-closed sets?

Answer 1

Given a supernatural number $N$, the set of positive integer divisors of $N$ is a strongly lcm-closed set. And any nonempty strongly lcm-closed set $X$ arises in this way, with $N$ equal to the supernatural lcm of the $x$ in $X$. (See Serre, Galois cohomology for the notion of supernatural number: it is a formal product over primes, $\prod_p p^{n_p}$, where each $n_p$ is in $\lbrace 0,1,2, \dots,\infty \rbrace$.)

Answer 2

Those would be the ideals in the lattice of positive integers ordered by divisibility.