TIL about directed sets: sets with a preorder \geq and the additional property that for any pair of elements (a, b) an element c exists such that c \geq a and c \geq b.

In a directed set, every maximal element (m is maximal if no x > m exists) is a greatest element (m is greatest if for every x it holds that m \geq x).

Linearizations (and chunkings) of graphs/clusters are directed sets (with \gtrsim as the direction). The merge-intersection algorithm computes the c mentioned above given a and b. The “move highest feerate subset to front, continue with remainder” algorithm by construction computes a maximal element, which is thus a greatest element. “Optimal linearization” is the name we have for greatest element.