Perfect monoids with zero and categories of S-acts

Abstract

In this paper, we study the relationship between the two main categories of S-acts for a monoid S with zero from the viewpoint of existence of projective covers. In particular, we prove that the condition that all acts have a projective cover holds in the category of all acts if and only if it holds in the category of all pointed acts. Furthermore, all connected pointed acts are cyclic if and only if they satisfy the ascending chain condition on cyclic subacts.

Keywords:

• Act over a monoid
• perfect monoid
• steady monoid

2020 Mathematics Subject Classification:

• 20M50 (20M30)

Acknowledgments

The authors thank the referee for his or her careful reading of the paper and many suggestions leading to an improvement of the paper.