Abstract
In this paper, we propose the concept of an [Formula: see text]-Rota–Baxter system, which is a generalization of a Rota–Baxter system and an [Formula: see text]-Rota–Baxter algebra of weight zero. In the framework of operated algebras, we obtain a linear basis of a free [Formula: see text]-Rota–Baxter system for an extended diassociative semigroup [Formula: see text], 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 [Formula: see text]-Rota–Baxter systems, and construct their free objects. Meanwhile, free [Formula: see text]-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.
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