## Determining set membership from ordering relationships among disjoint sets

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.

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Your problem is a little unclear at the end. In particular, you need to spell out more clearly what the inputs to the question are. As written now, it seems to me that part of the initial data you're given are explicit elements $p_i\in P$. But then it seems that you should know right off the bat which partition $P_i$ each element is in.