K-additive set-valued maps on a cylinder

Let $$(X,+)$$ be a group and $$(Y,+)$$ be a commutative monoid. We prove that the Cauchy nucleus of a set-valued map $$F:X\rightarrow 2^Y\setminus \{\emptyset \},$$ i.e. the set $$\{y\in X:\,F(x+y)=_KF(x)+F(y)\;\text{ for } \text{ every }\;x\in X\},$$ is a subgroup of X provided it is nonempty, and that every subgroup of X is the Cauchy nucleus of a set-valued map $$F:X\rightarrow 2^Y\setminus \{\emptyset \}$$ . We apply these results to characterize solutions of a partially Pexiderized Cauchy equation $$F(x+y)=_KF(x)+G(y)$$ for every $$(x,y)\in X\times S$$ , and also solutions of this equation satisfied almost everywhere (in the sense of an ideal) in $$X\times S$$ , provided S is a subgroup of X.