On Cogenerating and Finitely cogenerated S-acts

TL;DR

This paper introduces and characterizes cogenerating and finitely cogenerated S-acts, exploring their relationships with other classes like cofaithful and subgenerator acts, and analyzing their roles in generating injective envelopes.

Contribution

It defines and characterizes cogenerating and finitely cogenerated S-acts, and introduces new classes like cofaithful and subgenerator S-acts, expanding the understanding of S-act structures.

Findings

Finitely cogenerated S-acts have socles that are finitely cogenerated and large.

Identified classes of cofaithful and subgenerator S-acts as dual notions of faithful acts.

Established relationships among various classes of S-acts, including cogenerating and generating classes.

Abstract

In this paper, cogenerating classes of $S$-acts are introduced as those that can be used to cogenerate $S$-acts in an appropriate sense. Next, finitely cogenerated $S$-acts are introduced. Such an $S$-act is then characterized as one whose socle is finitely cogenerated and large in it. Further, we investigate the $S$-acts cogenerating $S_S$, or generating the injective envelope $E(S)$ of $S_S$. This leads us to introduce the classes of cofaithful and subgenerator $S$-acts as the dual notions of faithful $S$-acts, lying strictly between the classes of generator and faithful $S$-acts. Ultimately, we investigate the relationship between the mentioned classes of $S$-acts.