Simple characterization of Fubini-Bruhat covering relations

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

Background

The paper introduces three related partial orders on Fubini words: the espresso order, the medium roast order, and the decaf order. It establishes several families of covering relations, including the Transposition, Pushback, and Superpushback rules, and proves a lifting property for the medium roast and touching relations.

Despite these results, the paper states that a simple description of the covering relations remains unresolved. Thus, the open problem concerns a complete and conceptually simple characterization rather than the particular covering families proved in the paper.