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In this question it was asked if the division lattice (i.e., the preorder category $(\Bbb Z_{>0}, \mid)$) contains enough information categorically to recover the relation $ab=\gcd(a,b)\operatorname{lcm}(a,b)$.

I answered that this is impossible because the division lattice does not know what multiplication is; we first have to introduce it as a tensor product.

However, while it seemed intuitively obvious that this tensor product cannot be recovered from the division lattice, I wasn't sure of how to prove this.

I am interested in both general techniques to prove this kind of things impossible and a proof for the specific situation.

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Lord_Farin
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1 Answers1

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I contend that we can reconstruct $\mathbb Z_{>0}$ up to a permutation of the form $$\prod_i p_i^{a_i}\mapsto \prod_i \pi(p_i)^{a_i}.$$ Since such maps are homomorphisms of the multiplicative monoid we can reconstruct multiplication.

We can associate with every $x$ the set of proper divisors, i.e., $D(x):=\{\,y\in\mathbb Z_{>0}:y\ne x, y\mid x\,\}$.

We identify $1$ as the only object $x$ with $D(x)=\emptyset$.

Next we identify the set of primes as $$\mathbb P=\{\,x\in\mathbb Z_{>0}:D(x)=\{1\}\,\}.$$

Given $q\in \mathbb P$ we can recursively identify $q^n$ as the unique object $x$ with $D(x)=\{\,q^i:0\le i<n\,\}$.

Given $a\in\mathbb Z_{>0}$ and a prime $q$ we can determine $v_q(a)$ as the maximal $n$ such that $q^n\mid a$.

Finally, given $a,b\in\mathbb Z_{>0}$ we can identify $ab$ as the unique object $x$ with $v_q(x)=v_q(a)+v_q(b)$ for all primes $q$.

Hagen von Eitzen
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