On Some Generalizations Of Pure Epimorphism And Unitary Property Of S-acts

Let S be a monoid and n be a positive integer. Let ψ:Xâ†' Y be an S-epimorphism in which X satisfies condition (P). In 2014, Bailey and Renshaw proved if ψ is 2-pure then Y satisfies condition (P), but the converse is not true. In this paper, we give some new generalizations of pure epimorphism. Using these new concepts, we obtain some sufficient and necessary characterizations of condition (E’), condition (PWP), condition (P), condition (P’), condition (PE), condition (EP), condition (E’P) and we also investigate directed colimits of these new epimorphisms.