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 unital bialgebra structure is defined on the vector space H PP generated by PP , using the notion of biideals of plane posets. e 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 nondegenerate Hopf pairing on the infinitesimal unital bialgebra H PP .