Some Results on Homomorphisms of Hypergroups

A homomorphism of a hypergroup (H, ◦) is a function f : H → H satisfying f(x ◦ y) ⊆ f(x) ◦ f(y) for all x, y ∈ H. A homomorphism of a hypergroup (H, ◦) is called an epimorphism if f(H) = H. Denote by Hom(H, ◦) and Epi(H, ◦) the set of all homomorphisms and the set of all epimorphisms of a hypergroup (H, ◦), respectively. In this paper, the elements of Hom(Zn, ◦m) and Epi(Zn, ◦m) are characterized where (Zn, ◦m) is the hypergroup Zn with the hyperoperation ◦m defined by x ◦m y = x + y + mZn for all x, y ∈ Z. In addition, |Hom(Zn, ◦m)| and |Epi(Zn, ◦m)| are determined. Mathematics Subject Classification: 20N20