On Extensions of Set-Valued Homomorphisms “modulo K”

Let X be a group, Y be a monoid, K be a submonoid of Y and \(n(Y):=2^Y\setminus \{\emptyset \}\) $n(Y):=2^Y\backslash \{\mathrm{\varnothing }\}$ . In the paper we prove that under some additional assumptions set-valued K-homomorphism \(G:S\rightarrow n(Y)\) $G:S\to n(Y)$ defined on a subgroup S of X can be extended to a set-valued K-homomorphism \(F:X\rightarrow n(Y)\) $F:X\to n(Y)$ .