Algebraic structures on double and plane posets

TL;DR

This paper explores the algebraic properties of Hopf algebras related to double and plane posets, revealing their free, cofree, and self-dual structures, and connecting them to operads of 2-As and B_infty algebras.

Contribution

It introduces new algebraic structures and explicit pairings for Hopf algebras of double and plane posets, and characterizes their operads in terms of plane posets.

Findings

Hopf algebras of plane posets are free, cofree, and self-dual

Explicit Hopf pairing constructed for these algebras

Description of operads of 2-As and B_infty algebras via plane posets

Abstract

We study the Hopf algebra of double posets and two of its Hopf subalgebras, the Hopf algebras of plane posets and of posets "without N". We prove that they are free, cofree, self-dual, and we give an explicit Hopf pairing on these Hopf algebras. We also prove that they are free 2-As algebras; in particular, the Hopf algebra of posets "without N" is the free 2-As algebra on one generator. We deduce a description of the operads of 2-As algebras and of B_infty algebras in terms of plane posets.