Let $P$ be a poset and denote by $Hom(P, \mathbb N)$ the set of all monotone functions from $P$ to natural numbers $\mathbb N$. Under what conditions on $P$ Is it possible to recover the order on $P$ from the knowledge of $Hom(P, \mathbb N)$?

I should mention here that the only example I am interested in is the poset of prime ideals in a commutative Noetherian ring.

It would be great if you could include references.

Thanks!

Let $P$ be a poset and denote by $Hom(P, \mathbb N)$ the set of all monotone functions from $P$ to $\mathbb N$. Under what conditions on $P$ Is it possible to recover the order on $P$ from the knowledge of $Hom(P, \mathbb N)$?

I should mention here that the only example I am interested in is the poset of prime ideals in a commutative Noetherian ring.

It would be great if you could include references.

Thanks!

Let $P$ be a poset and denote by $Hom(P, \mathbb N)$ the set of all monotone functions from $P$ to $\mathbb N$. Under what conditions on $P$ Is it possible to recover the order on $P$ from the knowledge of $Hom(P, \mathbb N)$?

It would be great if you could include references.

Thanks!