The Jordan-Hölder Theorem for Monoids with Group Action

In this article, we prove an isomorphism theorem for the case of refinement $Γ$-monoids. Based on this we show a version of the well-known Jordan-Hölder theorem in this framework. The main theorem of this article states that - as in the case of modules - a monoid $T$ has a $Γ$-composition series if and only if it is both $Γ$-Noetherian and $Γ$-Artinian. As in module theory, these two concepts can be defined via ascending and descending chains respectively.

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