Free commutative matching tridendriform algebras and universal enveloping matching Rota–Baxter algebras

In this paper, we introduce the concept of a weighted matching tridendriform algebra, which generalizes both the tridendriform algebras introduced by Loday and Ronco and the [Formula: see text]-tridendriform algebras proposed by Burgunder and Ronco. Furthermore, we construct a free commutative matching tridendriform algebra using the matching augmented shuffle product. Finally, we introduce the concept of universal enveloping commutative matching Rota–Baxter algebras of commutative matching (tri)dendriform algebras and give a construction of the universal enveloping algebras.