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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.

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?

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Excellent! This is exactly what I'm after. By the way, I would accept both answers below, if I could. Thanks! – Douglas S. Stones Feb 21 '10 at 21:51
up vote 5 down vote accepted

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 $\lbrace 0,1,2, \dots,\infty \rbrace$.)

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I do not see why every strongly-lcm set could be obtained this way, how about {p^2a} for p fix? – domotorp Feb 21 '10 at 14:43
Your example doesn't satisfy the definition: we have lcm(p,p^2) in X, so p and p^2 should be in X too. – Bjorn Poonen Feb 21 '10 at 15:26
Oh, you are right. – domotorp Feb 21 '10 at 20:27

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

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