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Let a cluster be a term with a number of patterns occurring in it. We givetwo accounts of clusters, a geometric one as sets of (node and edge) positions,and an inductive one as pairs of terms with gaps (2nd order variables) andpattern-substitutions for the gaps. We show both notions of cluster and thecorresponding refinement/coarsening orders on them, to be isomorphic. Thisequips clusters with a lattice structure which we lift to (parallel/multi)steps to yield an alternative account of the notion of critical peak.

Critical Peaks Redefined - $\Phi \sqcup \Psi = \top$