The Hopf algebra of integer binary relations

TL;DR

This paper introduces a Hopf algebra framework on integer binary relations that unifies several existing algebras related to permutahedra and associahedra, and also derives new algebraic structures on related combinatorial objects.

Contribution

It constructs a comprehensive Hopf algebra encompassing multiple known structures and introduces new Hopf structures on intervals of permutation and binary tree orders.

Findings

Unifies known Hopf algebras within a single framework

Derives new Hopf structures on permutation and binary tree intervals

Connects algebraic structures to combinatorial objects like permutahedra and associahedra

Abstract

We construct a Hopf algebra on integer binary relations that contains under the same roof several well-known Hopf algebras related to the permutahedra and the associahedra: the Malvenuto-Reutenauer algebra on permutations, the Loday-Ronco algebra on planar binary trees, and the Chapoton algebras on ordered partitions and on Schr\"oder trees. We also derive from our construction new Hopf structures on intervals of the weak order on permutations and of the Tamari order on binary trees.