On Rees weakly projective right acts

In this article, we present an extension of the concept of weakly projective acts to the so-called $(S/I,S/J)$-projective acts. Retracts, coproducts, and products of acts as well as Rees factor acts are considered from the point of view of these properties. They are used to describe monoids that are disjoint unions of a group with a left zero semigroup or with a disjoint union of simple right ideals. We suggest concepts of weak QF-monoids.