θ*-Relation on Hypermodules and Fundamental Modules Over Commutative Fundamental Rings

Anvariyeh, S. M.; Mirvakili, S.; Davvaz, B.

doi:10.1080/00927870701724078

Cited by 30 publications

(8 citation statements)

References 11 publications

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“…Then, this relation is investigated in [59]. Also, a similar relation is defined on hypermodules to obtain an ordinary module [13,14,58]. The largest class of hyperstructures called H v -structures.…”

Section: Fundamental Relations On Hyperstructuresmentioning

confidence: 99%

A brief survey on algebraic hyperstructures: Theory and applications

Davvaz

2020

JAHLA

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I am working on algebraic hyperstructures from 1995. During the last twenty years, I together with my students and co-authors studied and developed the theory of algebraic hyperstructures in many directions. In particular, we tried to find real examples of hyperstructures in nature. In this paper we review some parts of these works such as (1) Fundamental relations on hyperstructures; (2) Fuzzy sets and hyperstructures; (3) Rough sets and hyperstructures; (4) Topology and hyperstructures; (5) Number theory and hyperstructures; (6) n-ary hypergroups and there extension to hyperrings and hypermodules; (7) Applications of hyperstructures in biology, physics and chemistry.

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Section: Fundamental Relations On Hyperstructuresmentioning

confidence: 99%

A brief survey on algebraic hyperstructures: Theory and applications

Davvaz

2020

JAHLA

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“…Then θ * is a strongly regular relation both on (M, +) and M as an R-hypermodule. Also the (abelian group) M/θ * is an R/α * -module, where R/α * is a commutative ring and the relation θ * is the smallest equivalence relation such that the (abelian) quotient M/θ * is an R/α * -module [1].…”

Section: θ -Relation On Hypermodulesmentioning

confidence: 99%

Strongly transitive geometric spaces associated to hypermodules

Anvariyeh

Davvaz

2009

Journal of Algebra

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In this paper, we use the strongly regular θ * -relation on hypermodules (with canonical hypergroup) over a given Krasner hyperring. In this way, we consider the fundamental relation θ * defined on a hypermodule and prove some results in this respect. Also, we determine a family P σ (M) of subsets of a hypermodule M and we give sufficient conditions such that the geometric space (M, P σ (M)) is strongly transitive and the relation θ is transitive.

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“…Besides, a commutative ring can be obtained as a quotient structure of a hyperring modulo the α * -relation [12]. Clearly, when we want to link hyperstructures with structures by this method of factorization via a strongly regular relation, we need to consider particular structures, as nilpotent groups [1], engel groups [2], solvable groups [15], Boolean rings [10], commutative rings with identity [3], commutative modules [5].…”

Section: Introductionmentioning

confidence: 99%

The commutative quotient structure of m-idempotent hyperrings

Zadeh

Norouzi

Cristea

2020

Analele Universitatii "Ovidius" Constanta - Seria Matematica

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The α* -relation is a fundamental relation on hyperrings, being the smallest strongly regular relation on hyperrings such that the quotient structure R/α* is a commutative ring. In this paper we introduce on hyperrings the relation ζm, which is smaller than α*, and show that, on a particular class of m-idempotent hyperrings R, it is the smallest strongly regular relation such that the quotient ring R/ζ*m is commutative. Some properties of this new relation and its differences from the α* -relation are illustrated and discussed. Finally, we show that ζm is a new representation for α* on this particular class of m-idempotent hyperrings.

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θ*-Relation on Hypermodules and Fundamental Modules Over Commutative Fundamental Rings

Cited by 30 publications

References 11 publications

A brief survey on algebraic hyperstructures: Theory and applications

A brief survey on algebraic hyperstructures: Theory and applications

Strongly transitive geometric spaces associated to hypermodules

The commutative quotient structure of m-idempotent hyperrings

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