Abstract

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 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 nondegenerate Hopf pairing on the infinitesimal unital bialgebra HPP.

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