Right projective semigroups with 0

We describe the semigroups S with 0 that are projective in the category of right S-objects, i.e. of centered right S-operands. Among them are the free semigroups with an adjoined zero, and several characterizations of this subclass are given. Two other classes, one of them close to free semigroups with 0, the other rather far apart, are discussed in detail. The first class consists of semigroups of finite and infinite sequences, with multiplication based on concatenation. The second is formed by the projectives in the category of (S,S)-biobjects.