On Homomorphisms of Krasner Hyperrings

Abstract

On Homomorphisms of Krasner Hyperrings By a homomorphism from a Krasner hyperring (A, +, ·) into a Krasner hyperring (A', +', ·') we mean a function ƒ: A → A' satisfying ƒ(x + y) ⊆ ƒ(x)+ ƒ(y) and ƒ(x · y) = ƒ(x) ·' ƒ(y) for all ×, y ∈ A. The kernel of ƒ, ker ƒ, is defined by ker ƒ = {x ∈ A | ƒ(x) = 0'} where 0' is the zero of (A', +', ·'). In fact, ker ƒ may be empty. In this paper, some general properties of a Krasner hyperring homomorphism with nonempty kernel are given. Various examples are also provided.

Topological Krasner hyperrings with special emphasis on isomorphism theorems

“…a 11 a 12 a 13 a 14 a 21 a 22 a 23 a 24 a 31 a 32 a 33 a 34 a 41 a 42 a 43 a 44     be an element of M 4 ([0, 1]) and con- Proof. Clearly, ψ : H 1 / ker f → H 2 defined by ψ(x + ker f ) = f (x), for all x ∈ H 1 is the required isomorphism [24,7]. So, it is only required to prove that ψ is open and continuous.…”

“…Hyperring, introduced by Krasner [16] is one of the most general structures so far in the literature that satisfies the ring-like axioms. Later, many mathematicians, like Ameri [3,2], Massouros [18], Spartalis [29], Davvaz [7], Stratigopoulos [30], Kemprasit [24] extended this field of study. In literature, a topological ring is a combination of two structures, namely a topological space and a ring.…”

“…A homomorphism from a hyperring (H, +, [24]) that the kernel of a homomorphism may be empty, but, if it is nonempty (i.e., ker f = φ), then the following results ( [24]) hold:…”

The associated hyperringoid to a Krasner hyperring

“…Example 1.16: Let F 2 = {0, 1} be the finite set with two elements. Then F 2 becomes a Krasner hyperfield with the following hyperoperation + and binary operation [29].…”

Composition hyperrings

“…A Krasner hyperring is a nonempty set R endowed with a hyperoperation (the addition) and a binary operation (the multiplication) such that (R, +) is a canonical hypergroup, (R, ·) is a semigroup and the multiplication is distributive with respect to the addition. The theory of these hyperrings has been developing since the beginning of seventies, thanks to the contributions of Mittas [14,15], Krasner [10], Stratigopoulos [20], till nowadays [2,3,5,8,13,17].…”