On the abstract elementary class of acts with embeddings

TL;DR

This paper explores acts with embeddings as an abstract elementary class, establishing stability, characterizing superstability algebraically, and linking weakly noetherian monoids to model-theoretic properties and pure acts.

Contribution

It introduces parametrized weakly noetherian monoids, characterizes them via injective acts, and extends classical ring theory results to acts.

Findings

The class of acts with embeddings is always stable.

Superstability is characterized algebraically by weakly noetherian monoids.

Weakly noetherian monoids are characterized via absolutely pure acts.

Abstract

We study the class of acts with embeddings as an abstract elementary class. We show that the class is always stable and show that superstability in the class is characterized algebraically via weakly noetherian monoids. The study of these model-theoretic notions and limit models lead us to introduce parametized weakly noetherian monoids and find a characterization of them via parametrized injective acts. Furthermore, we obtain a characterization of weakly noetherian monoids via absolutely pure acts extending a classical result of ring theory. The paper is aimed at algebraists and model theorists so an effort was made to provide the background for both.