Rota-Baxter family systems and Gr\"{o}bner-Shirshov bases

Abstract:

      Motivated by the concept of Rota-Baxter family algebras arising from associative Yang-Baxter family equations and Volterra integral equations, we introduce the notion of a Rota-Baxter family system which generalizes the Rota-Baxter system proposed by Brzezi$\acute{\mathrm{n}}$ski. We show that this notation is also related to an associative Yang-Baxter family pair and the pre-Lie family algebras. Furthermore, as an analogue of Rota-Baxter family system, we introduce a notion of averaging family system and prove that an averaging family system induces a dialgebra family structure. We also study Rota-Baxter family systems on a dendriform algebra and show how they induce quadri family algebra structures. Finally, we give a linear basis of the Rota-Baxter family system by the methods of Gr\"obner-Shirshov bases.