Non-commutative Arithmetics on Thompson's Monoid  

作　　者：Bo Qing 

机构地区：[1]XUE Institute of Mathematical Sciences,ShanghaiTech University,Shanghai 201210,P.R.China

出　　处：《Acta Mathematica Sinica,English Series》2021年第10期1586-1626,共41页数学学报（英文版）

基　　金：Supported by National Natural Science Foundation of China(Grant No.11701549)。

摘　　要：The main purpose of this paper is to define prime and introduce non-commutative arithmetics based on Thompson's group F.Defining primes in a non-abelian monoid is highly non-trivial,which relies on a concept called"castling".Three types of castlings are essential to grasp the arithmetics.The divisor functionτon Thompson's monoid S satisfiesτ(uv)≤τ(u)τ(v)for any u,v∈S.Then the limitτ_(0)(u)=lim_(n→∞)(τ(u~n))^(1/n)exists.The quantityC(S)=sup_(1≠u∈S)τ_(0)(u)/τ(u)describes the complexity for castlings in S.We show thatC(S)=1.Moreover,the MCbius function on S is calculated.And the Liouville functionCon S is studied.

关 键 词：Thompson's group non-commutative arithmetics castling COMPLEXITY divisor function M?bius function 

分 类 号：O152.7[理学—数学]