Abstract
We generalize the concept of weakly regularity in semigroups to S-acts,
where S is a monoid. We prove among other results that if a monoid
is von-Neumann regular then weakly regularity and von-Neumann regularity, in the context of S-acts, coincide. We also define locally projective S-acts, which is the generalization of projective S-acts. We
consider many relationships between weakly regular S-acts and locally
projective S-acts.