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 PPPP of isoclasses of plane posets owns two products, and an infinitesimal unital bialgebra structure is defined on the vector space HPPHPP generated by PPPP, using the notion of biideals of plane posets.We here define a partial order on PPPP, making it isomorphic to the set of partitions with the weak Bruhat order. We prove that this order is compatible with both products of PPPP; moreover, it encodes a nondegenerate Hopf pairing on the infinitesimal unital bialgebra HPPHPP.