I am trying to figure out a good way of finding the intersection of two partitioned subsets of a set (or what to call what I'm trying to do so I can read something about it).
Let's say I have two subsets, ABC and BCDE. The first is partitioned into A/BC (part A vs part BC) and the second is partitioned into B/CDE (part B vs part CDE). Whatever the two subsets are, the partitions are always bipartite.
Now I want to combine these subsets, retaining the partitions so that I have A/B/C/DE. I may implicitly be treating the fact of the two subsets as a higher-order partition. This seems to me some sort of intersection of partitions (hence title).
It seems there should be a clear algorithm for what I'm doing here but I just can't grasp it. To make things worse, I need to be able to scale this up to more than two subsets (but the partitions are always bipartite). Any advice would be appreciated.
Thanks for reading--
You can equivalently think of partitions as equivalence relations, with elements in the same piece of the partition labelled as equivalent. So the partition in the first subset $S_1$ is given by the equivalence $$B \sim_1 C,$$ and the partition in the second subset $S_2$ is given by the equivalence $$C \sim_2 D \sim_2 E.$$ We can even suppose that we have more subsets, $S_1,...,S_n$, each with a partition defined by the equivalence relation $\sim_n$. In order to match your operation, it's actually better if we extend these relations to the union $\bigcup_1^n S_i$ by additionally declaring $X \sim_i Y$ if $X, Y \notin S_i$.
Now I define a final equivalence relation $\sim$ by $X \sim Y \iff X \sim_i Y \;\forall\; i$. Translating this back into a partition should match the operation you've defined.