Structure theorems of mixable shuffle algebras and free commutative Rota-Baxter algebras

TL;DR

This paper investigates the algebraic structures of mixable shuffle and free commutative Rota-Baxter algebras, establishing structure theorems by leveraging connections with shuffle algebras, Lyndon words, and quasi-symmetric functions.

Contribution

It provides new structure theorems for a broad class of mixable shuffle and Rota-Baxter algebras using combinatorial and algebraic methods.

Findings

Established structure theorems for mixable shuffle algebras

Connected mixable shuffle algebras with Lyndon words and shuffle products

Applied results to various coefficient rings

Abstract

We study the ring theoretical structures of mixable shuffle algebras and their associated free commutative Rota-Baxter algebras. For this study we utilize the connection of the mixable shuffle algebras with the overlapping shuffle algebra of Hazewinkel, quasi-shuffle algebras of Hoffman and quasi-symmetric functions. This connection allows us to apply methods and results on shuffle products and Lyndon words on ordered sets. As a result, we obtain structure theorems for a large class of mixable shuffle algebras and free commutative Rota-Baxter algebras with various coefficient rings.