Typed Angularly Decorated Planar Rooted Trees and Ω-Rota-Baxter Algebras

As a generalization of Rota–Baxter algebras, the concept of an $\mathsf{\Omega }$-Rota–Baxter could also be regarded as an algebraic abstraction of the integral analysis. In this paper, we introduce the concept of an $\mathsf{\Omega }$-dendriform algebra and show the relationship between $\mathsf{\Omega }$-Rota–Baxter algebras and $\mathsf{\Omega }$-dendriform algebras. Then, we provide a multiplication recursion definition of typed, angularly decorated rooted trees. Finally, we construct the free $\mathsf{\Omega }$-Rota–Baxter algebra by typed, angularly decorated rooted trees.