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In this paper, we propose the concept of an Ω -Rota–Baxter system, which is a generalization of a Rota–Baxter system and an Ω -Rota–Baxter algebra of weight zero. In the framework of operated algebras, we obtain a linear basis of a free Ω -Rota–Baxter system for an extended diassociative semigroup Ω , in terms of bracketed words and the method of Gröbner–Shirshov bases. As applications, we introduce the concepts of Rota–Baxter system family algebras and matching Rota–Baxter systems as special cases of Ω -Rota–Baxter systems, and construct their free objects. Meanwhile, free Ω -Rota–Baxter algebras of weight zero, free Rota–Baxter systems, free Rota–Baxter family algebras and free matching Rota–Baxter algebras are reconstructed via new method. [ABSTRACT FROM AUTHOR]

In this paper, we first introduce a weighted derivation on algebras over an operad 풫, and prove that for the free 풫-algebra, its weighted derivation is determined by the restriction on the generators. As applications, we propose the concept of weighted differential (q-tri)dendriform algebras and study some basic properties of them. Then Novikov-(tri)dendriform algebras are initiated, which can be induced from differential (q-tri)dendriform of weight zero. Finally, the corresponding free objects are constructed, in both the commutative and noncommutative contexts. [ABSTRACT FROM AUTHOR]

In this paper, we obtain, respectively, some new linear bases of free nonunitary (modified) weighted differential algebras and free nonunitary (modified) Rota–Baxter algebras, in terms of the method of Gröbner–Shirshov bases. [ABSTRACT FROM AUTHOR]