The Jordan-H\"older Theorem for Monoids with Group Action

TL;DR

This paper extends the Jordan-H"older theorem to refinement $ ext{Gamma}$-monoids, establishing conditions for the existence of $ ext{Gamma}$-composition series based on $ ext{Gamma}$-Noetherian and $ ext{Gamma}$-Artinian properties.

Contribution

It proves an isomorphism theorem for refinement $ ext{Gamma}$-monoids and establishes a Jordan-H"older type theorem in this algebraic framework.

Findings

Monoids with group action can have a Jordan-H"older theorem analogue.

A monoid has a $ ext{Gamma}$-composition series iff it is $ ext{Gamma}$-Noetherian and $ ext{Gamma}$-Artinian.

The main theorem parallels module theory results in the context of $ ext{Gamma}$-monoids.

Abstract

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.