Bruhat order on plane posets and applications

A plane poset is a finite set with two partial orders, satisfying a certain incompatibility condition. The set PP of isoclasses of plane posets owns two products, and an infinitesimal Hopf algebra structure is defined on the vector space HPP generated by PP , using the notion of biideals of plane posets. We here define a partial order on PP , making it isomorphic to the set of partitions with the weak Bruhat order. We prove that this order is compatible with both products of PP ; moreover, it encodes a non degenerate Hopf pairing on the infinitesimal Hopf algebra HPP .