Refinement of a set partition

A set partition $\mathcal{Q}$ refines $\mathcal{P}$ if every cell of $\mathcal{Q}$ is contained in a cell of $\mathcal{P}$. Thus $\mathcal{Q}$ retains every separation made by $\mathcal{P}$ and may make additional separations. Refinement of a set partition is reflexive and transitive. If each partition refines the other, their cells coincide, so refinement of a set partition is a partial order on partitions. For finite partitions the nonempty intersections of cells form their common refinement of a set partition. In graph regularity arguments, subdividing cells and treating discarded vertices as singletons preserves refinement of a set partition; merging discarded vertices into one cell need not. The refinement variance identity for regularity energy consequently proves that this singleton treatment cannot decrease the equitable regularity energy.