Retractable and Coretractable Acts Over Semigroups

Let S be a monoid. A right S-act A is said to be retractable if for each subact B of A, Hom(A, B) 6 not equal phi. In this paper we define an S-act A to be coretractable if Hom(A/B, A) 6 not equal {0}, for every proper subact B of A. Our objective is to characterize retractable and coretractable acts by properties of underlying sets and monoids and investigate the relation between these properties of acts with some "projective or injective likes" properties. Clearly every generator is retractable and every cogenerator is coretractable in the category of S-acts. It turns out that for a monoid S with a left zero all free acts are coretractable if and only if Hom(A, S) not equal {0}, for each S-act A.