Porcupine-quotient graphs, the fourth primary color, and graded composition series of Leavitt path algebras

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 $L_K(E)\mathrm{/}I$ is graded isomorphic to the Leavitt path algebra of the {\em quotient graph} of $(H,S)\mathrm{.}$ 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} $(G,T)\mathrm{/}(H,S)$ such that its Leavitt path algebra is graded isomorphic to the quotient $I(G,T)\mathrm{/}I(H,S)\mathrm{.}$ 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 $M_E^{\mathrm{\Gamma }}\mathrm{.}$ An ideal of $M_E^{\mathrm{\Gamma }}$ is minimal exactly when it is generated by the element of $M_E^{\mathrm{\Gamma }}$ corresponding to a terminal vertex.