2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/99154The quasi-shuffle product and mixable shuffle product are both generalizations of the shuffle product and have both been studied quite extensively recently. We relate these two generalizations and realize quasi-shuffle product algebras as subalgebras of mixable shuffle product algebras. As an application, we obtain Hopf algebra structures in free Rota-Baxter algebras.14 pages, no figure, references updatedRings and AlgebrasCommutative AlgebraCombinatoricsQuantum Algebra16A06, 05E99, 16W30text