A note on near-rings and hypernear-rings with a defect of distributivity

Jančić-Rašović, Sanja; Cristea, Irina

doi:10.1063/1.5043950

Cited by 3 publications

(5 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Another important property of division near-rings says that the additive part of a division near-ring is commutative [25]. This is one open problem in hypernear-rings theory, that we intend to investigate in our future research, and also to extend it to the case of general hypernear-rings [15].…”

Section: Discussionmentioning

confidence: 99%

See 1 more Smart Citation

Division hypernear-rings

Jančić-Rašović

Cristea

2018

Analele Universitatii "Ovidius" Constanta - Seria Matematica

Self Cite

View full text Add to dashboard Cite

This paper is a continuation of our work on hypernear-rings with a defect of distributivity D. In particular, here we introduce and study a new subclass of hypernear-rings, called D-division hypernear-rings, establishing a necessary and sufficient condition such that a hypernear-ring with the defect D is a D-division hypernear-ring. Several properties and examples of these two subclasses of hypernear-rings are presented and discussed.

show abstract

Section: Discussionmentioning

confidence: 99%

“…for any theree elelments x, y, z, we have x • (y + z) = x • y + x • z, then we call the hypernear-ring a strongly distributive hypernear-ring. If the additive structure is a hypergroup and all the other properties related to the multiplication are conserved, we obtain a general hypernear-ring [15].…”

Section: Introductionmentioning

confidence: 99%

Division hypernear-rings

Jančić-Rašović

Cristea

2018

Analele Universitatii "Ovidius" Constanta - Seria Matematica

Self Cite

View full text Add to dashboard Cite

show abstract

“…For further properties of these concepts we refer the reader to the papers [2,3,11,12] and the fundamental books [13][14][15]. For the consistence of our study, regarding hypernear-rings we keep the terminology established and explained in [8,16].…”

Section: Preliminariesmentioning

confidence: 99%

“…Then the structure (R, ⊕ P 1 , P 2 ) is a general left hypernear-ring [8,18]. Let H = R ∪ {a} and define on H the hyperoperation ⊕ P as follows:…”

Section: Remark 1 Let (G +) Be a Group And T(g) Be The Transformations Near-ring On G Obviously T(g) ⊂mentioning

confidence: 99%

“…Later on, this algebraic hyperstructure was called a strongly distributive hypernear-ring, or a zero-symmetric hypernear-ring, while in a hypernear-ring the distributivity property was replaced by the "inclusive distributivity" from the left (or right) side. Moreover, when the additive part is a hypergroup and all the other properties related to the multiplication are conserved, we talk about a general hypernear-ring [8]. The distributivity property is important also in other types of hyperstructures, see e.g., [9].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Weak Embeddable Hypernear-Rings

2019

Self Cite

View full text Add to dashboard Cite

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.

show abstract