Weak Embeddable Hypernear-Rings

Abstract

In this paper we extend one of the main problems of near-rings to the framework of algebraic hypercompositional structures. This problem states that every near-ring is isomorphic with a near-ring of the transformations of a group. First we endow the set of all multitransformations of a hypergroup (not necessarily abelian) with a general hypernear-ring structure, called the multitransformation general hypernear-ring associated with a hypergroup. Then we show that any hypernear-ring can be weakly embedded into a multitransformation general hypernear-ring, generalizing the similar classical theorem on near-rings. Several properties of hypernear-rings related with this property are discussed and illustrated also by examples.

Superring of Polynomials over a Hyperring

“…In the following, by some counterexamples, we will show that the Theorem 3.2 in [34] is not true because, in the hyperstructure of polynomials over a Krasner hyperring, the hypermultiplication is not strongly distributive with respect to the hyperaddition, even if we replace a hyperring with a hyperfield. In the following, we will show that the polynomial over a hyperring(or a hyperfield) constitutes a superring, which is called the superring of polynomials.…”

“…(i) If in (iii) of the above definition the equality holds, then R is called an strongly distributive superring. (ii) Every strongly distributive superring R is in fact an additive-multiplicative hyperring in the sense [34]. 1. for every a, b ∈ A implies a − b ⊆ A; 2. for every a ∈ A, r ∈ S implies r.a ⊆ A(resp.…”

“…Next, lemma is a superring version of Lemma 3.1 in [34] Lemma 1. Let S be a superring and X ⊂ S. Then, for a ∈ S, the following statements are satisfied:…”

Derived Hyperstructures from Hyperconics

“…M. Krasner introduced the concept of the hyperfield and hyperring in Algebra [20,21]. The theory which was developed for the hyperrings is generalizing and extending the ring theory [22][23][24][25]. There are different types of hyperrings [22,25,26].…”

“…The theory which was developed for the hyperrings is generalizing and extending the ring theory [22][23][24][25]. There are different types of hyperrings [22,25,26]. In the most general case a triplet (R, +, •) is a hyperring if (R, +) is a hypergroup, (R, •) is a semihypergroup and the multiplication is bilaterally distributive with regards to the addition [3].…”