Fig. 1

Partition lattice for a set of four elements, a, b, c, d. This Hasse diagram shows all possible ways of clustering elements into subsets, organized by refinement ordering. In a partition lattice, each node represents a distinct partition of the set, with edges directed from coarser partitions (fewer subsets) to finer partitions (more subsets). The coarsest partition at the top groups all elements together as a, b, c, d, while the finest partition at the bottom separates each element individually as a, b, c, d. The VI distance is aligned along this lattice, with the lowet VI distances corresponding to the smallest refinement