Coalgebras, bialgebras and Rota-Baxter algebras from shuffles of rooted forests

TL;DR

This paper develops new algebraic structures on rooted trees and forests using shuffle operations, creating bialgebras and Rota-Baxter algebra connections, advancing the mathematical understanding of these combinatorial objects.

Contribution

It introduces novel coproducts and products on rooted trees and forests, establishing bialgebra and Rota-Baxter algebra structures from shuffle operations.

Findings

Constructed a coproduct on rooted trees forming a bialgebra.

Characterized the dual coproduct to the shuffle product of rooted forests.

Proved properties of modified shuffles in Rota-Baxter algebra category.

Abstract

We construct and study new generalisations to rooted trees and forests of some properties of shuffles of words. First, we build a coproduct on rooted trees which, together with their shuffle, endow them with bialgebra structure. We then caracterize the coproduct dual to the shuffle product of rooted forests and build a product on rooted trees to obtain the bialgebra dual to the shuffle bialgebra. We then characterize and enumerate primitive trees for the dual coproduct. Finally, using modified shuffles of rooted forests, we prove a property in the category of Rota-Baxter algebras.