Strongly Hopfian and Co-Hopfian Acts over Monoids: Structure and Characterizations
Ali Madanshekaf, Farideh Farsad
TL;DR
The paper extends endomorphism-based stability concepts from modules to $S$-acts over a monoid by introducing strongly Hopfian and strongly co-Hopfian acts. It analyzes the relationships between these strong notions and classical properties such as Noetherian, Artinian, injective, projective, quasi-injective, and quasi-projective, providing equivalences and structural results (e.g., stabilization of the chains $\mathcal{K}_f$ and $\mathcal{I}_f$ for every endomorphism $f$). A key result shows that under suitable conditions, a quasi-projective (resp. quasi-injective) $S$-act that is strongly co-Hopfian (resp. strongly Hopfian) is also strongly Hopfian (resp. strongly co-Hopfian). The paper also presents numerous examples and subquotient behavior results for strongly Hopfian and strongly co-Hopfian acts, deepening the understanding of endomorphism-dynamics in the category Act-$S$. These findings broaden the structural toolbox for monoid actions and may inform further study of endomorphism-induced invariants in act categories.
Abstract
In this paper, we introduce and explore new classes of S-acts over a monoid S, namely, strongly Hopfian and strongly co-Hopfian acts, as well as their weaker counterparts, Hopfian and co-Hopfian acts. We investigate the relationships between these newly defined structures and well-studied classes of S-acts, including Noetherian, Artinian, injective, projective, quasi-injective, and quasi-projective acts. A key result shows that, under certain conditions, a quasi-projective (respectively quasi-injective) S-act that is strongly co-Hopfian (respectively strongly Hopfian) is also strongly Hopfian (respectively strongly co-Hopfian). Moreover, we provide a variety of examples and structural results concerning the behavior of subacts and quotient acts of strongly Hopfian and strongly co-Hopfian S-acts, further elucidating the internal structure and interrelationships within this extended framework.