Characterizing posets by functions into natural numbers 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.
 A: 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).
A: If you have given all linear extensions of $\mathcal{L}(P)$ of a poset $P$. This is the set of all linear orderings (permutations) of the vertex set of $P$ preserving the order in $P$. The order in $P$ can than be recovered as $v < w$ in P iff $v <_\sigma w$ for all permutations $\sigma \in \mathcal{L}(P)$.
Proof:
One direction is obvious: if $v < w$ in P, then, by definition, $v <_\sigma w$ for all permutations $\sigma \in \mathcal{L}(P)$.
To obtain the other direction, observe that posets are in 1-1 correspondence with directed acyclic graphs, see e.g. the last paragraph of http://en.wikipedia.org/wiki/Partially_ordered_set#Strict_and_non-strict_partial_orders.
Let us consider $P$ to be identified with its directed acyclic graph. Since adding an edge either from $v$ to $w$ or from $w$ to $v$ does not create a cycle in this graph, we have posets $P_{v < w}$ and $P_{w < v}$ with $v < w$ and $w < v$ respectively. Since
$$ \emptyset \neq \mathcal{L}(P_{v < w}), \mathcal{L}(P_{w < v}) \subseteq \mathcal{L}(P),$$
we finally found two permutations $\sigma,\tau \in \mathcal{L}(P)$ with $v < _ \sigma w$ and $w < _ \tau v$. $\qquad\square$
For references see the wiki page on linear extensions:
http://en.wikipedia.org/wiki/Linear_extension
Or am I misunderstanding something in your question?
Best, Christian
A: If I'm not mistaken, Novotny's paper "Ueber gewisse Eigenschaften von Kardinaloperationen" deals with [and I think solves, if you use the partial ordering on $Hom(P,\mathbb N)$] this question.
