Abstract: Let S  be a monoid,A  is quasi-projective S -system if for every S -epimophism f:A→B and S-homomorphism g:A→B,there exists a S-homomorphism h:A→A such that fh=g.The properties of quasi-projective S -system are studied and two main results are obtained:(1) S is a(semi) perfect monoid if and only if for every (f.g.) right S-system have a quasi-projective cover S -system;(2) S is a right hereditary monoid if and only if every projective S -systemis quasi-projective.

Key words: projective system, quasi-projective system, (semi)perfect monoids, hereditary monoids