The Jordan–Hölder theorem for monoids with group action

Abstract

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.