Structure Theorems of Mixable Shuffle Algebras

Mixable shuffle algebras are generalizations of the well-known shuffle algebra and quasi-shuffle algebra with broad applications. In this article we study the ring theoretic structures of mixable shuffle algebras with coefficients in a field motivated by the well-known work of Radford that a shuffle algebra with rational coefficients is a polynomial algebra in Lyndon words. To consider coefficients in a field of positive characteristic p, we carefully study the Lyndon words and their p-variations. As a result, we determine the structures of a quite large class of mixable shuffle algebras by providing explicit sets of generators and relations.