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In this paper, we prove an isomorphism theorem for the case of refinement monoids with a group [Formula: see text] acting on it. Based on this, we show a version of the well-known Jordan–Hölder theorem in this framework. The central result of this paper states that — as in the case of modules — a monoid [Formula: see text] has a [Formula: see text]-composition series if and only if it is both [Formula: see text]-Noetherian and [Formula: see text]-Artinian. As in module theory, these two concepts can be defined via ascending and descending chains, respectively.

We characterise directed graphs consisting of disjoint cycles via their talented monoids. We show that a graph E consists of disjoint cycles precisely when its talented monoid T E has a certain Jordan-Hölder composition series. These are graphs whose associated Leavitt path algebras have finite Gelfand-Kirillov dimension. We show that this dimension can be determined as the length of certain ideal series of the talented monoid. Since T E is the positive cone of the graded Grothendieck group K gr 0 (L k (E)), we conclude that for graphs E and F , if, thus providing more evidence for the Graded Classification Conjecture for Leavitt path algebras.

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In this article, we study properties as simplicity, solvability and nilpotency for Lie bracket algebras arising from Leavitt path algebras, based on the talented monoid of the underlying graph. We show that graded simplicity and simplicity of the Leavitt path algebra can be connected via the Lie bracket algebra. Moreover, we use the Gelfand-Kirillov dimension for the Leavitt path algebra for a classification of nilpotency and solvability.