  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.
