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 lcmx ∈ Xx. 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.



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This question arose in my [Ph.D. thesis](https://sites.google.com/site/rknmodn/papers) (p.107) where strongly lcm-closed sets describe where autotopisms of Latin squares give rise to subsquares.


As a side question:



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> Is there a common name for strongly lcm-closed sets?
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Classification of strongly lcm-closed sets
Given a [supernatural number](http://en.wikipedia.org/wiki/Supernatural_numbers) $N$, 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 $\lbrace 0,1,2, \dots,\infty \rbrace$.)


