令$Γ$是生成元$x.$上的无限循环群为了避免在处理具有额外$\mathbb Z$ -作用的$\mathbb Z$ -模块时产生混淆，我们将$\mathbb Z$ -作用视为$Γ$ -作用。从一个有向图$E$出发，可以定义一个可消交换幺半群$M_E^Γ$，它的$Γ$ -作用与幺半群的结构和自然序一致.顺序和动作使人们能够将每个非零元素标记为以下之一：可比（周期或非周期）或不可比。我们全面配对这些元素的功能与图论性质的元素的发电机。我们还刻画了图，使每个元素的$M_E^Γ$是可比的，周期性的，图形，使每个非零元素的$M_E^Γ$是非周期性的，不可比的，图形，使没有非零元素的$M_E^Γ$是周期性的，和图形，使没有元素的$M_E^Γ$是非周期性的。分次分类猜想可以表述为$M_E^Γ$是域$K.$上$E$的Leavitt路代数$L_K(E)$的完全不变量。我们的刻画表明，由于$E$的性质很好地反映在$M_E^Γ.$的结构上，分次分类猜想可能有一个肯定的答案。哈兹拉特Li，The talented monoid of a Leavitt path algebra，J. Algebra，547（2020）430-455]在不要求图是行有限的情况下成立。
英文摘要：
Let $Γ$ be the infinite cyclic group on a generator $x.$ To avoid confusion when working with $\mathbb Z$-modules which also have an additional $\mathbb Z$-action, we consider the $\mathbb Z$-action to be a $Γ$-action instead.
Starting from a directed graph $E$, one can define a cancellative commutative monoid $M_E^Γ$ with a $Γ$-action which agrees with the monoid structure and a natural order. The order and the action enable one to label each nonzero element as being exactly one of the following: comparable (periodic or aperiodic) or incomparable. We comprehensively pair up these element features with the graph-theoretic properties of the generators of the element. We also characterize graphs such that every element of $M_E^Γ$ is comparable, periodic, graphs such that every nonzero element of $M_E^Γ$ is aperiodic, incomparable, graphs such that no nonzero element of $M_E^Γ$ is periodic, and graphs such that no element of $M_E^Γ$ is aperiodic.
The Graded Classification Conjecture can be formulated to state that $M_E^Γ$ is a complete invariant of the Leavitt path algebra $L_K(E)$ of $E$ over a field $K.$ Our characterizations indicate that the Graded Classification Conjecture may have a positive answer since the properties of $E$ are well reflected by the structure of $M_E^Γ.$ Our work also implies that some results of [R. Hazrat, H. Li, The talented monoid of a Leavitt path algebra, J. Algebra, 547 (2020) 430-455] hold without requiring the graph to be row-finite.
Some more talents of the talented monoid of a higher-rank graph
Western Sydney University（西悉尼大学）；Center for Pure Mathematics, School of Mathematical Sciences, Anhui University（安徽大学数学科学学院纯数学中心）；Department of Mathematics, Jadavpur University（贾达普大学数学系）
Super adaptable graphs and the realization problem
Universitat Autònoma de Barcelona（巴塞罗那自治大学）；Universidad de Zaragoza（萨拉戈萨大学）；Universidad de Cádiz（加的斯大学）；Universitat Politècnica de Catalunya - BarcelonaTech (UPC)（加泰罗尼亚理工大学）
Properties of the $\mathcal V$-Monoid of Weighted Leavitt Path Algebras
North-Eastern Hill University（东北丘陵大学）；Research Center for Theoretical Physics, Central Visayan Institute Foundation（中维萨亚斯研究所理论物理研究中心）；Linnaeus University（林奈大学）；Central Mindanao University（棉兰老中央大学）
