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 calle...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.展开更多
基金Supported by National Natural Science Foundation of China(Grant No.11701549)。
文摘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.
