Homomorphisms of Some P-hyperrings

By a hyperring we mean a structure (A, ⊕, ◦) where (A, ⊕ )i s ah ypergroup, (A, ◦) is a semihypergroup and x ◦ (y ⊕ z) ⊆ x ◦ y ⊕ x ◦ z and (y ⊕ z) ◦ x ⊆ y ◦ x ⊕ z ◦ x for all x,y,z ∈ A .I f (R,+, ·) is a ring and P1,P2 are nonempty subsets of R satisfying RP2P1 ∪ P1P2R ⊆ P1, then (R, ⊕P1, ◦P2) is a hyperring where x ⊕P1 y = x + y + P1 and x ◦P2 y = xP2y for all x,y ∈ R. The hyperring (R, ⊕P1 , ◦P2 ) is called a P-hyperring. A function f : A → A is called a homomorphism of (A, ⊕, ◦ )i ff(x ⊕ y) ⊆ f(x) ⊕ f(y) and f(x ◦ y) ⊆ f(x) ◦ f(y) for all x,y ∈ A. The purpose of this paper is to show that for the ring (Zn,+, ·),Hom(Zn,+) ⊆ Hom(Zn, ⊕l n , ◦m n ) if and only if n (m,n) is square-free where Hom(Zn,+) and Hom(Zn, ⊕l n , ◦m n ) are the set of all homomorphisms of the group (Zn,+) and the set of all homomorphisms of the hyperring (Zn, ⊕l n, ◦m n), respectively.