论文标题
图中图中的可比性
Comparability in the graph monoid
论文作者
论文摘要
让$γ$为发电机$x。$上的无限环状群，以避免使用$ \ m athbb z $ -Modules，它也具有额外的$ \ mathbb z $ -Action，我们认为$ \ Mathbb z $ -Action是$γ$ -Action。 从有向图的$ e $开始，可以定义取消交换性的单体$ m_e^γ$，并具有$γ$ - action，这与单体结构和自然顺序相符。该顺序和动作使每个非零元素都标记为以下一个：可比较（周期性或上隔离）或无与伦比。我们将这些元素特征与元素发电机的图理论属性进行了全面结合。我们还表征了图表，使得$ m_e^γ$的每个元素都是可比性的，周期性的图形，使每个非零元素的$ m_e^γ$的每个非零元素都是aperiodic的，无与伦比的图形，因此没有$ m_e^γ$的非零元素是周期性的，并且没有元素的元素。 可以表明，分级分类的猜想可以指出，$ m_e^γ$是laevitt路径的完全不变的代数$ l_k $ l_k（e）$ e $ a $ e $的$ e $。 Hazrat，H。Li，Leavitt Path代数，J。代数，547（2020）430-455的才华横溢的MONOID，而无需该图为行。
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.
