= 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>.
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