Completely quasi-projective monoids (Q1102375)

Let S be a monoid with a two-sided zero element. A right S-system M is quasi-projective if, for a given S-epimorphism \(\mu\) : \(M\to A\) and an S- homomorphism \(f: M\to A\), there exists an S-homomorphism \(g: M\to M\) such that \(\mu g=f\). S is completely quasi-projective if each right S-system is quasi-projective. In this paper we show that if S is such a monoid then \(S=\{1,0\}\). If S is also commutative then S is completely quasi- projective if and only if each coproduct of quasi-projective S-system is quasi-projective (if and only if each quasi-projective S-system is projective).