Hi,


Here are a couple of questions:


1. Is there a way to classify all homomorphisms between two finite posets?
2. Same question as (1) but for infinite, locally finite countable connected posets.


These questions are relevant to causal set quantization.
Thank you


PS: I mention a partial result concerning AUTOMORPHISMS of locally finite countable connected posets: any automorphism of such a poset (assuming some extra hypotheses) should either transform chains into themselves (not all chains, but a set of chains forming a partition of the poset), or tranform antichains into themselves (again a partition of the poset). So in the first case the automorphism is induced by automorphisms of a chain (which can be indexed in a clear way by integers), and in the second case the automorphism is induced by automorphisms of an antichain (that, too, satisfy further conditions).
So a similar characterization in terms of homomorphisms is helpful.


For the meaning of homomorphism we can require that if $a \leq b$ then $\pi(a) \leq \pi(b)$ and if $\pi(a) \leq \pi(b)$ then there exists $a'$ and $b'$ such that $a' \leq b'$ and $\pi(a')=\pi(a)$ and $\pi(b') =\pi(b)$. 



Causal sets  quantization
I think you should study rather endomorphisms of posets than homomorphisms. Then you obtain an algebraic structure (the semigroup of endomorphisms) and in addition a lot of articles in this problem. See, e.g., 
P. M. Higgins, J. D. Mitchell, M. Morayne and N. Ruškuc, Rank Properties of Endomorphisms of Infinite Partially Ordered Sets, Bull. London Math. Soc. (2006) 38 (2): 177-191.


