Preordered forests, packed words and contraction algebras

TL;DR

This paper introduces generalized forest structures called preordered and heap-preordered forests, proves their algebraic properties as Hopf algebras, and constructs morphisms to packed words and shuffle algebras.

Contribution

It extends the theory of ordered forests by defining preordered variants, establishing their Hopf algebra structures, and connecting them to packed words and shuffle algebras.

Findings

Preordered and heap-preordered forests form Hopf algebras under the cut coproduct.

Constructed a Hopf morphism from these forests to the Hopf algebra of packed words.

Defined a new coproduct via edge contraction and described morphisms to shuffle and quasi-shuffle algebras.

Abstract

We introduce the notions of preordered and heap-preordered forests, generalizing the construction of ordered and heap-ordered forests. We prove that the algebras of preordered and heap-preordered forests are Hopf for the cut coproduct, and we construct a Hopf morphism to the Hopf algebra of packed words. Moreover, we define another coproduct on the preordered forests given by the contraction of edges. Finally, we give a combinatorial description of morphims defined on Hopf algebras of forests with values in the Hopf algebras of shuffes or quasi-shuffles.