The Jordan–Hölder theorem for monoids with group action

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