Words Transformation Of Artin-Tits Groups

This paper mainly discusses the special transformation of words in positive presen-tation,including 0,1,2,oo transformation,Dehn transformation and reverse transfor-mation of words.The thesis is divided into three chapters.The first chapter,introduction,mainly introduces the research background and main results of this paper.In the second chapter,the transformation of 0,1,2,? of a word in a free semigroup is introduced,and the property H is introduced.Then it introduces the Dehn trans-formation of the word,explores the relationship between the 0,D transformation and the 0,1,2transformation,and gives the Artin-Tits group presentation that the word representing 1 in the group cannot carry out the 0,Dtransformation.The third Chapter mainly introduces word inversion.Firstly,some definitions and conclusions of word inversion are introduced.Then the relation between word equivalence and word inversion is explored,and an example is given that word equivalence cannot be detected by word inversion.However,they are equivalent under complete conditions,and we give a method to complete incomplete presentation.Finally,the solvable formula method of the word problem is given when the complete presentation satisfies certain conditions,and the solvable word problem of the finite type right Angle Artin-Tits group is proved.Some examples of finite Artin-Tits groups are also given to illustrate the solvability of the existing word problem,and finally gave the proof.