Homomorphisms of some hypergroups

Abstract

A homomorphism of a hypergroup (H,) is a function f : H→H such that f (x y) f (x)  f ( y) for all x, y H . If the equality holds, f is called a good homomorphism of (H,) . A homomorphism f of a hypergroup (H,) is called an epimorphism if f (H) = H . For a hypergroup (H,) , denote by Hom (H,) , GHom (H,) , Epi (H,) and GEp (H,) the set of all homomorphisms, the set of all good homomorphisms, the set of all epimorphisms and the set of all good epimorphisms of (H,) , respectively. If G is a group and N is a normal subgroup of G , let (G, ) be the hypergroup where the hyperoperation N is defined by x y = xyN for all x, y G, The elements of GHom (Z, mZ) and (Z, mZ) have been characterized. It was also shown that | GHom (Z, Z) | = | GEpi (Z,Z) | = 2 N0 if m ≠ 0 . The main purpose of this research is to characterize the elements of Hom (Z, Z) , Epi (Z,Z) , Hom (Zn, ),GHom (Zn, ), Epi (Zn, ) and GEpi (Zn, ). In addition, the cardinalities of these sets are given. This research also includes some results on homomorphisms of the following hypergroups : P -hy-pergroups, hypergroups defined from abelian groups whose hyperproducts are subgroups and the hypergroup defined from R whose hyperproducts are closed intervals.