Multiple Linear Operators,rewriting Systems And Gr(?)bner-Shirshov Bases

Linear operators have a significant role in mathematic research by working on research objects,such as examples of differential operators in differential algebra(as an algebraization of differential operators in analysis)and Rota operators in Rota-Baxter algebra(as an algebraization of integral operators in analysis).Inspired by this,Rota,as a famous mathematician for combination,posed an open problem as follows: finding all possible algebraic identities that can be satisfied by a linear operator on an algebra,which was called Rota's classification problem.Then Professor Li Guo and others did an accurate description by using Gr (?)bner-Shirshov bases theory and methods of rewriting systems,and verified a cluster of operated polynomial identity are Gr (?)bner-Shirshov bases if and only if the rewriting systems they induce are convergent,where the linear operator is a linear operator.In this paper,we continue to study Rota's classification problem,generalizing some results proved by professor Li Guo about multiple linear operators.We discuss multiple linear operators by using of Gr (?)bner-Shirshov bases theory and methods of rewriting systems,and verify a cluster of multiple linear operated polynomial identity are Gr (?)bner-Shirshov bases if and only if the rewriting systems they induce are convergent.As an application,we verify a rewriting system is convergent,which is constructed by defined as a weight of differential Rota-Baxter linear operator,then obtain these identities the Gr (?)bner-Shirshov bases.This paper has three chapters.The first chapter introduces multiple linear operator origin,development and its status in algebra: the theory of ?-algebra has an important developing potential in the theory of algebra,as a subject between ring theory and universal algebra.In addition,we state briefly main ideas,the most important problem solved and construction in this paper.The chapter two reviews concepts of free operated monoid,free operated algebra,direct system and direct limit and so on.Then we introduce definitions of multiple linear operated polynomial identities,operated ideals and rewriting systems induced from multiple linear identities.We learn concepts of rewriting system and its basic properties,and introduce definitions of leading monomial,monomial order,intersection composition and including composition of some identities,with some examples for better understanding.Lastly we discuss Gr (?)bner-Shirshov bases and Composition-Diamond lemma.In the last chapter,we revisit application of Newman lemma on rewriting systems and term rewriting system properties of confluent and terminating.Then we discuss the relationship between Gr (?)bner-Shirshov bases and convergent rewriting systems based on multiple linear operator identities.We prove that a family of multiple linear operator identities are Gr (?)bner-Shirshov bases if and only if rewriting systems they induce are convergent.In addition,we verify a differential Rota-Baxter rewriting system is confluent,which is induced by the double linear operated identities defining the differential RotaBaxter algebra of weight ,and obtain these identities are the Gr (?)bner-Shirshov bases.