AI 中文总结
本文研究高阶图的有才幺半群的性质，证明其可捕捉高阶图的关键几何信息，分次K-理论可区分两类Kumjian-Pask代数，还推导了高阶图代数纯无限单等的有才幺半群判据。
AI 中文摘要
本文探讨了分次Grothendieck群$K_0^{gr}$，即其正锥有才幺半群，可检测高阶图代数（即高阶图$C^*$代数和Kumjian-Pask代数）的结构类型这一思路。我们证明有才幺半群能捕捉高阶图的部分关键几何信息，包括带入口和不带入口的循环的存在性；进而证明分次K-理论可有效区分局部有限Kumjian-Pask代数类与交叉积Kumjian-Pask代数类。我们还推导了高阶图代数为纯无限单、非AF或非超矩阵代数的有才幺半群判据。
英文摘要
In this paper, we explore the idea that the graded Grothendieck group $K_0^{gr}$, or equivalently its positive cone, the talented monoid, can detect the structural type of higher-rank graph algebras (i.e., higher-rank graph $C^*$-algebras and Kumjian--Pask algebras). We show that the talented monoid captures some of the essential geometric information of a higher-rank graph, including the existence of cycles with and without entrances. In turn, we show that the graded $K$-theory can effectively distinguish the class of locally finite Kumjian--Pask algebras, and also the class of crossed product Kumjian--Pask algebras. We also derive talented monoid criteria for higher-rank graph algebras to be purely infinite simple, and not to be $AF$ or ultramatricial.
