Name of this commutative monoid over the divisors of a integer

Question

Let $D_n$ be the set of divisors of $n$, and define a operation $\cdot$, such that $x,y\in D_n$, $x\cdot y=\gcd(xy,n)$.

$(D_n,\cdot)$ is a commutative monoid. Is there a more well known name for such structure?

Answer 1

Your monoid is related to the way little chidren count: $0, 1, 2, 3, 4, 5, \ldots,$ many $\ldots$

More precisely, the threshold $t$ monoid is the monoid $\Bbb N_t = \{0, 1, 2, \ldots, t\}$ under the addition $\oplus$ defined by $x \oplus y = \min(x+y, t)$. Let now $$ n = \prod_{p \text{ prime}}p^{n_p} $$ be the prime decomposition of $n$. Then $D_n$ is isomorphic to the product of the monoids $\Bbb N_{n_p}$ such that $n_p \not = 0$.

To answer your question, I don't know of a specific name for $D_n$, but the monoid of divisors of $n$ seems to be appropriate (you should of course define the product).