scholarly journals Transitivity of the εm-relation on (m-idempotent) hyperrings

2018 ◽  
Vol 16 (1) ◽  
pp. 1012-1021 ◽  
Author(s):  
Morteza Norouzi ◽  
Irina Cristea
Keyword(s):  
Equivalence Relation ◽  
Transitive Closure ◽  
Fundamental Relation ◽  
Strongly Regular ◽  
Regular Equivalence ◽  
Quotient Set ◽  
Classical Ring

AbstractOn a general hyperring, there is a fundamental relation, denoted γ*, such that the quotient set is a classical ring. In a previous paper, the authors defined the relation εm on general hyperrings, proving that its transitive closure $\begin{array}{} \displaystyle \varepsilon^{*}_{m} \end{array}$ is a strongly regular equivalence relation smaller than the γ*-relation on some classes of hyperrings, such that the associated quotient structure modulo $\begin{array}{} \displaystyle \varepsilon^{*}_{m} \end{array}$ is an ordinary ring. Thus, on such hyperrings, $\begin{array}{} \displaystyle \varepsilon^{*}_{m} \end{array}$ is a fundamental relation. In this paper, we discuss the transitivity conditions of the εm-relation on hyperrings and m-idempotent hyperrings.

Γ∗-Relation on Fuzzy Hyperrings and Fundamental Ring

2021 ◽  
pp. 1-11
Author(s):  
N. Firouzkouhi ◽  
B. Davvaz
Keyword(s):  
Universal Algebra ◽  
Equivalence Relation ◽  
Fundamental Relation ◽  
Complete Part ◽  
Strongly Regular ◽  
Regular Equivalence ◽  
Fundamental Ring ◽  
Algebraic Hyperstructure

Fundamental relation performs an important role on fuzzy algebraic hyperstructure and is considered as the smallest equivalence relation such that the quotient is a universal algebra. In this paper, we introduce a new fuzzy strongly regular equivalence on fuzzy hyperrings such that the set of the quotient is a ring that is non-commutative. Also, we introduce the concept of a complete part of a fuzzy hyperring and study its principal traits. At last, we convey the relevance between the fundamental relation and complete parts of a fuzzy hyperring.

scholarly journals A Comparison of Complete Parts on m-Idempotent Hyperrings

Symmetry ◽  
2020 ◽  
Vol 12 (4) ◽  
pp. 554
Author(s):  
Azam Adineh Zadeh ◽  
Morteza Norouzi ◽  
Irina Cristea
Keyword(s):  
Equivalence Relation ◽  
Quotient Ring ◽  
Transitivity Property ◽  
Strongly Regular ◽  
Regular Equivalence ◽  
Symmetrical Group

On a particular class of m-idempotent hyperrings, the relation ξ m * is the smallest strongly regular equivalence such that the related quotient ring is commutative. Thus, on such hyperrings, ξ m * is a new representation for the α * -relation. In this paper, the ξ m -parts on hyperrings are defined and compared with complete parts, α -parts, and m-complete parts, as generalizations of complete parts in hyperrings. It is also shown how the ξ m -parts help us to study the transitivity property of the ξ m -relation. Finally, ξ m -complete hyperrings are introduced and studied, stressing on the fact that they can be characterized by ξ m -parts. The symmetry plays a fundamental role in this study, since the protagonist is an equivalence relation, defined using also the symmetrical group of permutations of order n.

scholarly journals Cyclic Groups Obtained as Quotient Hypergroups

2015 ◽  
Vol 61 (1) ◽  
pp. 109-122
Author(s):  
S.Sh. Mousavi ◽  
V. Leoreanu-Fotea ◽  
M. Jafarpour
Keyword(s):  
Cyclic Group ◽  
Equivalence Relation ◽  
Cyclic Groups ◽  
Strongly Regular ◽  
Regular Equivalence ◽  
Transitivity Condition

Abstract We introduce a strongly regular equivalence relation ρ*A on the hypergroup H, such that in a particular case the quotient is a cyclic group. Then by using the notion of ρ*A-parts, we investigate the transitivity condition of ρA. Finally, a characterization of the derived hypergroup Dc(H) has been considered.

Fundamental relations and identities of fuzzy hyperalgebras

2021 ◽  
pp. 1-10
Author(s):  
Narjes Firouzkouhi ◽  
Abbas Amini ◽  
Chun Cheng ◽  
Mehdi Soleymani ◽  
Bijan Davvaz
Keyword(s):  
Universal Algebra ◽  
Equivalence Relation ◽  
Polynomial Function ◽  
Equivalence Relations ◽  
Fundamental Relation ◽  
Term Function ◽  
Biological Phenomena ◽  
Definition Of

Inspired by fuzzy hyperalgebras and fuzzy polynomial function (term function), some homomorphism properties of fundamental relation on fuzzy hyperalgebras are conveyed. The obtained relations of fuzzy hyperalgebra are utilized for certain applications, i.e., biological phenomena and genetics along with some elucidatory examples presenting various aspects of fuzzy hyperalgebras. Then, by considering the definition of identities (weak and strong) as a class of fuzzy polynomial function, the smallest equivalence relation (fundamental relation) is obtained which is an important tool for fuzzy hyperalgebraic systems. Through the characterization of these equivalence relations of a fuzzy hyperalgebra, we assign the smallest equivalence relation α i 1 i 2 ∗ on a fuzzy hyperalgebra via identities where the factor hyperalgebra is a universal algebra. We extend and improve the identities on fuzzy hyperalgebras and characterize the smallest equivalence relation α J ∗ on the set of strong identities.

Maximal Order of an NG-group

2021 ◽  
pp. 33-38
Author(s):  
Faraj. A. Abdunabi
Keyword(s):  
Permutation Group ◽  
Equivalence Relation ◽  
Maximal Order ◽  
Quotient Set

This study was aimed to consider the NG-group that consisting of transformations on a nonempty set A has no bijection as its element. In addition, it tried to find the maximal order of these groups. It found the order of NG-group not greater than n. Our results proved by showing that any kind of NG-group in the theorem be isomorphic to a permutation group on a quotient set of A with respect to an equivalence relation on A. Keywords: NG-group; Permutation group; Equivalence relation; -subgroup

scholarly journals Generalized fuzzy hypergraphs and hypergroupoids

Filomat ◽  
2016 ◽  
Vol 30 (9) ◽  
pp. 2375-2387 ◽  
Author(s):  
Mahdi Farshi ◽  
Bijan Davvaz
Keyword(s):  
Equivalence Relation ◽  
Higher Order ◽  
Regular Equivalence ◽  
Fuzzy Hypergraphs

This article first generalizes the ordinary fuzzy hypergraphs to generalized fuzzy hypergraphs and it makes a connection between generalized fuzzy hypergraphs and fuzzy hyperstructures. We construct a partial fuzzy hypergroupoid associated with it, giving some properties of the associated fuzzy hyperstructure. Moreover, we construct higher order fuzzy hypergroupoids and study their properties. Finally, by considering a regular equivalence relation on a (g-f)p-hypergroupoid, we define a quotient (g-f)phypergroupoid and we investigate some relationships between diagonal product of hypergroupoids and p-product of (g-f)-hypergraphs.

On cyclic hypergroups

2019 ◽  
Vol 18 (11) ◽  
pp. 1950213
Author(s):  
Ze Gu
Keyword(s):  
Index Formula ◽  
Fundamental Relation ◽  
Strongly Regular ◽  
Single Power

In this paper, we introduce the concept of the index of a generator in a cyclic hypergroup, and show that a single power cyclic hypergroup is generated by an element with index [Formula: see text]. Also, a characterization of the fundamental relation on a cyclic hypergroup is given. Finally, we study corresponding quotient structures induced by regular (strongly regular) relations on cyclic hypergroups. As an application, the corresponding results on single power hypergroups are obtained.

Solvable groups derived from hypergroups

2016 ◽  
Vol 15 (04) ◽  
pp. 1650067 ◽  
Author(s):  
M. Jafarpour ◽  
H. Aghabozorgi ◽  
B. Davvaz
Keyword(s):  
Solvable Group ◽  
Equivalence Relation ◽  
Solvable Groups ◽  
Equivalence Classes ◽  
Strongly Regular

In this paper, we introduce the smallest equivalence relation [Formula: see text] on a hypergroup [Formula: see text] such that the quotient [Formula: see text], the set of all equivalence classes, is a solvable group. The characterization of solvable groups via strongly regular relations is investigated and several results on the topic are presented.

scholarly journals A further study on ordered regular equivalence relations in ordered semihypergroups

2018 ◽  
Vol 16 (1) ◽  
pp. 168-184 ◽  
Author(s):  
Jian Tang ◽  
Xinyang Feng ◽  
Bijan Davvaz ◽  
Xiang-Yun Xie
Keyword(s):  
Equivalence Relation ◽  
Open Problem ◽  
Equivalence Relations ◽  
Ordered Semigroups ◽  
Regular Equivalence

AbstractIn this paper, we study the ordered regular equivalence relations on ordered semihypergroups in detail. To begin with, we introduce the concept of weak pseudoorders on an ordered semihypergroup, and investigate several related properties. In particular, we construct an ordered regular equivalence relation on an ordered semihypergroup by a weak pseudoorder. As an application of the above result, we completely solve the open problem on ordered semihypergroups introduced in [B. Davvaz, P. Corsini and T. Changphas, Relationship between ordered semihypergroups and ordered semigroups by using pseuoorders, European J. Combinatorics 44 (2015), 208–217]. Furthermore, we establish the relationships between ordered regular equivalence relations and weak pseudoorders on an ordered semihypergroup, and give some homomorphism theorems of ordered semihypergroups, which are generalizations of similar results in ordered semigroups.

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