Porcupine-quotient graphs, the fourth primary color, and graded composition series of Leavitt path algebras
Lia Vas
TL;DR
The paper develops a porcupine-quotient framework for Leavitt path algebras, showing that graded ideals and their quotients are captured by admissible pairs and the porcupine-quotient graphs, with precise graded isomorphisms $L_K((G,T)/(H,S))\cong I(G,T)/I(H,S)$. It then links the existence of graded composition series to finite chains of admissible pairs whose consecutive porcupine-quotients are cofinal, and provides a constructive, checkable graph-theoretic criterion for such chains. A central contribution is the four-color (terminal-vertex) characterization of graded simple Leavitt path algebras, introducing terminal paths as a fourth color alongside sinks, cycles without exits, and extreme cycles. The work further connects these graph-theoretic insights to the graph monoid $M_E$ and the talented monoid $M_E^{\Gamma}$, detailing how minimal Gamma-order-ideals arise from terminal clusters and how cofinal porcupine-quotients yield monoid decompositions into periodic, aperiodic, or incomparable factors. Together, these results give a coherent, constructive approach to graded composition series for unital Leavitt path algebras and illuminate the structure of the associated monoids and their order-ideals.
Abstract
If $E$ is a directed graph, $K$ is a field, and $I$ is a graded ideal of the Leavitt path algebra $L_K(E),$ $I$ is completely determined by an admissible pair $(H,S)$ of two sets of vertices of $E$. The ideal $I=I(H,S)$ is graded isomorphic to the Leavitt path algebra of the {\em porcupine graph} of $(H,S)$ and the quotient $L_K(E)/I$ is graded isomorphic to the Leavitt path algebra of the {\em quotient graph} of $(H,S).$ We present a construction which generalizes both constructions and enables one to consider quotients of graded ideals: if $(H,S)$ and $(G,T)$ are admissible pairs such that $I(H,S)\subseteq I(G,T)$, we define the {\em porcupine-quotient graph} $(G,T)/(H,S)$ such that its Leavitt path algebra is graded isomorphic to the quotient $I(G,T)/I(H,S).$ Using the porcupine-quotient construction, the existence of a graded composition series of $L_K(E)$ is equivalent to the existence of a finite chain of admissible pairs of $E,$ starting with the trivial and ending with the improper pair, such that the quotient of two consecutive pairs is cofinal (a graph is cofinal exactly when its Leavitt path algebra is graded simple). We characterize the existence of such a composition series with a set of conditions which also provides an algorithm for obtaining such a series. The conditions are presented in terms of four types of vertices which are all ``terminal'' in a certain sense. Three types are often referred to as the three primary colors and the fourth type is new. As a corollary, a unital Leavitt path algebra has a graded composition series. We show that the existence of a composition series of $E$ is equivalent to the existence of a composition series of the graph monoid $M_E$ as well as a composition series of the talented monoid $M_E^Γ.$ An ideal of $M_E^Γ$ is minimal exactly when it is generated by the element of $M_E^Γ$ corresponding to a terminal vertex.