自然幺半群与非交换算术
Natural Monoids and Non-commutative Arithmetics
摘要：
我们介绍了几类幺半群满足多达五个公理，并建立了基本理论，他们的算术。满足所有公理的幺半群称为自然幺半群。两个典型的例子是：1）正有理数群中自然数的幺半群$\mathbb{N}$; 2）汤普森群中的某个幺半群$\mathbb{S}$。后者是非交换的，它是非交换算术的一个重要例子。在非阿贝尔幺半群$S$中定义素数是非常重要的，它依赖于我们称为“castling”的概念。要掌握$S$上的算术，三种类型的Castlings是必不可少的。定义了乘法函数和完全乘性函数。特别地，莫比乌斯函数是乘法的，而自然幺半群上的刘维尔函数是完全乘法的。除数函数具有次乘法性质，它在非阿贝尔幺半群$S$中诱导出非平凡量$τ_0(u)=\lim\nolimits_{n\rightarrow \infty}(τ(u^n))^{1/n}$。此外，数量$Ç(S)=\sup\nolimits_{1\neq u\in S}τ_0(u)/τ(u)$描述了$S$中castlings的复杂性。我们证明了$Ç(\mathbb{N})=1/2$和$Ç(\mathbb{S})=1$ .本文还研究了$S$的约化$C^\ast$ -代数，在其上可以定义一个特殊的迹.此外，我们证明了一个自然幺半群有许多素数是顺从的。
英文摘要：
We introduce several classes of monoids satisfying up to five axioms and establish basic theories on their arithmetics. The one satisfying all the axioms is named natural monoid. Two typical examples are 1) the monoid $\mathbb{N}$ of natural numbers in the group of positive rationals and 2) a certain monoid $\mathbb{S}$ in one of Thompson's groups. The latter one is non-abelian, which serves as an important example for non-commutative arithmetics. Defining primes in a non-abelian monoid $S$ is highly non-trivial, which relies on a concept we called `castling'. Three types of castlings are essential to grasp the arithmetics on $S$. Multiplicative and completely multiplicative functions are defined. In particular, Möbius function is multiplicative, and Liouville function on a natural monoid is completely multiplicative. The divisor function has a sub-multiplicative property, which induces a non-trivial quantity $τ_0(u)=\lim\nolimits_{n\rightarrow \infty}(τ(u^n))^{1/n}$ in a non-abelian monoid $S$. Moreover, the quantity $Ç(S)=\sup\nolimits_{1\neq u\in S}τ_0(u)/τ(u)$ describes the complexity for castlings in $S$. We show that $Ç(\mathbb{N})=1/2$ and $Ç(\mathbb{S})=1$. The reduced $C^\ast$-algebra of $S$, on which a particular trace can be defined, is also studied. Furthermore, we prove that a natural monoid having finitely many primes is amenable.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
