Rota-Baxter Family Systems and Gr\"{o}bner-Shirshov Bases
Received:March 03, 2022  Revised:August 19, 2022
Key Words: Rota-Baxter family systems   Rota-Baxter family algebras   Gr\"obner-Shirshov bases  
Fund Project:Supported by the National Natural Science Foundation of China (Grant No.12161013) and the Key Project of Guizhou University of Finance and Economics (Grant No.2022KYZD05).
Author NameAffiliation
Guo Shaungjian School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guizhou 550025, P. R. China 
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      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 notion 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.
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