Free $\Omega$-Rota-Baxter systems and Gr\"obner-Shirshov bases

In this paper, we propose the concept of an $\Omega$-Rota-Baxter system, which is a generalization of a Rota-Baxter system and an $\Omega$-Rota-Baxter algebra of weight zero. In the framework of operated algebras, we obtain a linear basis of a free $\Omega$-Rota-Baxter system for an extended diassociative semigroup $\Omega$, in terms of bracketed words and the method of Gr\"obner-Shirshov bases. As applications, we introduce the concepts of Rota-Baxter system family algebras and matching Rota-Baxter systems as special cases of $\Omega$-Rota-Baxter systems, and construct their free objects. Meanwhile, free $\Omega$-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.