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