Simple characterization of covering relations

Find a simple description of the covering relations in the espresso, medium roast, and decaf Fubini–Bruhat orders on Fubini words.

Background

The paper introduces three partial orders on Fubini words: the espresso order, the medium roast order, and the decaf order. It establishes several families of covering relations, including transposition, pushback, and superpushback rules, but these families do not provide a complete simple characterization.

The unresolved problem is to give a simple description of all covering relations for the Fubini–Bruhat orders, particularly in contrast with the well-understood covering relations in the classical Bruhat order on permutations.