Extensions of semigroups and morphisms of semigroup $C^*$-algebras

The paper is devoted to the normal extensions of discrete semigroups and $*$-homomorphisms of semigroup $C^*$-algebras. We study the normal extensions of abelian semigroups by arbitrary groups. Considering numerical semigroups, we prove that they are normal extensions of the semigroup of nonnegative integers by finite cyclic groups. On the other hand, we prove that if a semigroup is a normal extension of the semigroup of nonnegative integers by a finite cyclic group generated by a single element then this semigroup is isomorphic to a numerical semigroup. As regard a normal extension with a generating set, we consider two reduced semigroup $C^*$-algebras defined by this extension. We show that there exists an embedding of the semigroup $C^*$-algebras which is generated by an injective homomorphism of the semigroups and the natural isometric representations of these semigroups.