On the abstract elementary class of acts with embeddings
Marcos Mazari-Armida, Jiří Rosický
TL;DR
This work investigates the abstract elementary class of left $S$-acts with embeddings, aiming to connect model-theoretic dividing lines with semigroup-algebraic properties. The authors establish that the act-embedding AEC is always stable and prove a sharp algebraic criterion for superstability in terms of weakly left noetherian monoids, using limit-model techniques to bridge to parametrized notions. They introduce parametrized weakly noetherian monoids via parametrized injective acts and provide a characterization of weakly noetherian monoids through absolutely pure acts, extending classical results from ring theory to acts. The results extend the interaction between AECs and algebra beyond modules, offering new algebraic criteria and paving the way for broader categorical generalizations. Overall, the findings deepen the link between model-theoretic stability and finiteness-like algebraic conditions in semigroup theory, with potential applications to algebraic categories and first-order act theory.
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.