Rota–Baxter Operators and Systems on Hom-π-Algebras: Constructions, Derivation Chains, and Structural Equivalence

(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 monoid π. (2) Methods: We systematically investigate constructive approaches for Rota–Baxter operators on Hom-π-algebras, establish the stepwise structural derivation chain among Hom-tridendriform, Hom-dendriform, and Hom-associative π-algebras, and characterize the equivalence between generalized Rota–Baxter systems and dendriform structures in the Hom-π framework. (3) Results: We present several systematic constructive methods for Rota–Baxter Hom-π-algebras, including Yau twists, iterative endomorphism twisting, centroid element modifications, and monoid tensor lifting. We prove the structural equivalence between Rota–Baxter Hom-π-systems and Hom-dendriform π-algebras, and further extend the study to Hom-quadri-π-algebras with corresponding induction results. (4) Conclusions: The established structural equivalence and derivation relations provide a unified framework for these π-graded Hom-type algebraic structures, covering Rota–Baxter operators and various dendriform-derived algebras.