Homomorphisms of some hyperrings

Abstract

A homomorphism of a semihypergroup (H, )is a function ƒ : H → H such that ƒ(x ₀ y) ⊆ ƒ(x) ₀ ƒ( fy) for all x, y ϵ H . A homomorphism of a hyperring (A, ⊕, ₀) is a function ƒ : A → A such that ƒ is a homomorphism of both (A, ⊕) and (A, ₀) Denote by Hom (A, ⊕, ₀)the set of all homomorphisms of (A, ⊕, ₀) into itself. If (R, +, .)is a ring and I is an ideal of R, we write (R, +, ₀₁) for the multiplicative hyperring where x₀₁y = xy+I for all x, y ϵ R. The first purpose is to characterize when Hom(ℤ, +, ₀[subscript mℤ) = Hom(ℤ, +) and Hom(ℤ[subscript n], +, ₀[subscript mℤ[subscript n]]) = Hom(ℤ[subscript n], +) hold. We also show that Hom(ℤ, +, ₀[subscript mℤ) is infinite when m > 0, om(ℤ[subscript n], +, ₀ [subscript mℤ[subscript n]])| ≥ 2n/(m,n) when (m, n) > 1 and the equality holds if (m, n) is a prime power. We consider the two Krasner hyperrings (G⁰, ⊕₁, .) and (G⁰, ⊕₂, .) defined from a group (G, .) by G⁰ = G⊍{0}, 0⊕₁0 = {0}, x⊕₁0 = 0⊕₁x = {x}, x⊕₁x = G⁰\{x} for all x ϵ G, x⊕₁y = {x,y} for all distinct x,y ϵ G, 0⊕₂0 = {0}, x⊕₂0 = 0⊕₂x = {x}, x⊕₂x = {x, 0} สำหรับทุก x ϵ G, x⊕₂y = G\{x, y} for all distinct x, y ϵ G and 0.x = x.0 = 0 for all x ϵ G⁰. For the Krasner hyperring (G⁰, ⊕₂, .), the condition that | > 3 must be assumed. The second purpose is to characterize the elements of Hom(G⁰, ⊕₁, .) and Hom (G⁰, ⊕₂, .). The Krasner hyperring (R/ρ, ⊕, *) is considered where (R, +, .) is a commutative ring, x ρ y ⇔ x = y or x = -y, xρ ⊕ yρ = {(x+y)ρ, (x-y)ρ} and xρ*yρ = (xy)ρ for all x,y ϵ R. We characterize the elements of Hom(ℤ/ρ, ⊕, *) and ƒ ϵ Hom (ℤ[subscript n]/ρ, ⊕, *) with ƒ(0ρ) = 0ρ and ƒ(1ρ) = 1ρ. Moreover, the elements of Hom([0,∞), ⊕, .) are characterized where x⊕x = [0,x] for all xϵ[0,∞) and x⊕y = {max{x, y}} for all distinct x,y ϵ [0,∞) Let (R, ⊕[subscript P₁], ๐[subscript P₂]) be the P-hyperring of a ring (R, +, .) induced by nonempty subsets P₁, P₂ of R. The third purpose is to find Hom(ℤ, +)∩Hom(ℤ, ⊕[subscript lℤ], ๐[subscript mℤ]) and determine when Hom(ℤ[subscript n], +) is contained in Hom(ℤ[subscript n], ⊕[subscript lℤ], ๐[subscript mℤn]). The sets Hom(ℤ, ⊕[subscript lℤ], ๐([subscript mℤ])\Hom(ℤ, +) and Hom(ℤ[subscript n], ⊕[subscript lℤ], ๐[subscript mℤn])\Hom(ℤ[subscript n], +) are also shown to be nonempty for certain l, m.