Did you know that, if you have two partitions of a set, then they have a unique coarsest common refinement and a unique finest common coarsening, and that this gives the family of all possible partitions the structure of a lattice? And did you know that it's the kind of lattice that's secretly a matroid, and it's the kind of matroid that's secretly a graph? Specifically, it's a complete graph, the vertices of which form the set that's being partitioned (surprise! matroids are about edges but here it starts and ends with vertices), and each possible way of partitioning these vertices forms the set of connected components of a subgraph of the complete graph.
Finding connections like this, between concepts that I thought I understood but didn't realize were related to each other, is one of the things I like best about editing Wikipedia.
Some changes have been made to LiveJournal, and we hope you enjoy them! As we continue to improve the site on a daily basis to make your experience here better and faster, we would greatly appreciate your feedback about these changes. Please let us know what we can do for you!