Structural Interoperability between π-Graded Hom-Type Algebras and Rota-Baxter Operators

(1) Background: Hom-type algebras, proposed by Yau, generalize classical algebras via twisting maps. Dendriform and tridendriform algebras, introduced by Loday and Vallette, decompose associative multiplications and play significant roles in algebraic K-theory and operad theory. Rota-Baxter operators, originating from analysis and probability, have become a vital bridge connecting multiple disciplines. The π-graded structure, a classical tool in algebra, decomposes algebraic objects into direct sums indexed by a group π. (2) Methods: We systematically investigate the properties and construction methods of Rota-Baxter operators on π-graded Hom-algebras, and establish the derivation relations among π-graded Hom-tridendriform algebras, π-graded Hom-dendriform algebras and π-graded Hom-algebras. (3) Results: We prove that the generalized form, namely the π-graded Rota-Baxter system, is equivalent to π-graded Hom-dendriform algebras. Several iterative construction methods for π-graded Rota-Baxter Hom-algebras are also provided. (4) Conclusions: The structural equivalence between π-graded Rota-Baxter Hom-systems and π-graded Hom-dendriform algebras is established, providing a unified framework for these algebraic structures.