Adjoint to multiplication in a GCD lattice Consider the lattice on the nonzero natural numbers where the meet $a \wedge b $ is defined to be the greatest common divisor of $a$ and $b$, and the join $a \vee b$ is the least common multiple. There's a partial order in that $ a \leq b \equiv a \vee b = b$. With respect to that partial order, can we find an adjoint `/` to multiplication that satisfies the galois connection $ a/b \leq c \equiv a \leq b \times c$? With the ordinary notion of order on the naturals, we get residuated division as a _right_ adjoint to multiplication. But this doesn't work in the GCD lattice. Is there anything that does?

$a/b$ is just $a/\mathrm{gcd}(a,b)$. Here's a perversely highbrow explanation, just for fun.

Since the lattice of natural numbers under divisibility has meets of arbitrary nonempty sets (the greatest common divisor extends fine to infinite sets) any unbounded meet-preserving functor $f$ has a left adjoint $g$ given by $g(a)=\mathrm{inf}n:a \leq f(n)$. In the case that $f$ is multiplication by $b$, we have for $a/b$ the least $n$ such that $a$ divides $n\times b$, which is as described in the first paragraph.