Perfect monoids with zero and categories of $S$-acts

TL;DR

This paper explores the structure of categories of $S$-acts over monoids with zero, establishing conditions for projective covers and characterizing monoids where all compact acts are cyclic.

Contribution

It provides new characterizations of monoids with zero where all compact acts are cyclic and proves an equivalence between categories of $S$-acts based on projective covers.

Findings

Established an equivalence between categories of $S$-acts with projective covers

Characterized monoids with zeros where all compact acts are cyclic

Connected the existence of projective covers to the structure of $S$-act categories

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 and the equivalence is proven. Furthermore, monoids with zeros over which all compact acts are cyclic are characterized.