Let $P$ be a poset and denote by $\operatorname{Hom}(P, \mathbb N)$ 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 $\operatorname{Hom}(P, \mathbb N)$?


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.



Characterizing posets by functions into natural numbers
Sitting inside $Hom(P,{\bf N})$ is the set $P^\*$ of monotone functions from $P$ into $\{0,1\}$. This set carries the structure of a "Stone lattice", and the normal lattice homomorphisms from any Stone lattice into $\{0,1\}$ will be a poset, in the case of $P^\*$ it recovers $P$. In fact I have just described a dual equivalence between the category of posets with order preserving maps and the category of Stone lattices with normal lattice homomorphisms. See Theorem 6.33 of my book *Lipschitz Algebras* (second edition).


