Suppose we have a set $P$ (an infinite set), and we have a partition of $P$ into (finitely many) disjoint subsets $P_i$, so that $P = \cup_i P_i$, and $P_i \cap P_j = \emptyset$ for $i \neq j$.

Suppose now that we have a set of partial orderings of $P$, such that for each of the orderings, we know that they strictly order some of the partitions. I'm not sure if that is proper math-speak, but what I mean is, for example, for the first partial ordering given, it guarantees $a < b$ for all $a \in P_1$ and for all $b \in P_2$ (I will write this as $P_1 < P_2$). There may, for example, be another partial ordering given that guarantees $P_2 < P_1$.

My question is, given a finite set of $p_i \in P$ with a finite number of partitions $P_j$, as well as a finite set of partial orderings, is it possible to infer set membership of the $p_i$'s into the partitions ($P_j$'s)? If so, then what is required of the $p_i$ and the partial orderings? How many and which kinds of orderings need to be supplied?

Since I don't know the proper math terminology, I'm making this a wiki.