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10 pages, 265 KB

Open AccessArticle

Spectrum of Zariski Topology in Multiplication Krasner Hypermodules

by Ergül Türkmen, Burcu Nişancı Türkmen and Öznur Kulak

Mathematics 2023, 11(7), 1754; https://doi.org/10.3390/math11071754 - 6 Apr 2023

Viewed by 1500

In this paper, we define the concept of pseudo-prime subhypermodules of hypermodules as a generalization of the prime hyperideal of commutative hyperrings. In particular, we examine the spectrum of the Zariski topology, which we built on the element of the pseudo-prime subhypermodules of [...] Read more.

In this paper, we define the concept of pseudo-prime subhypermodules of hypermodules as a generalization of the prime hyperideal of commutative hyperrings. In particular, we examine the spectrum of the Zariski topology, which we built on the element of the pseudo-prime subhypermodules of a class of hypermodules. Moreover, we provide some relevant properties of the hypermodule in this topological hyperspace. Full article

(This article belongs to the Special Issue Topological Space and Its Applications)

7 pages, 227 KB

Open AccessArticle

A Study on k-Hyperideals in Ordered Semihyperrings

by Zheng Kou, Mehdi Gheisari and Saber Omidi

Symmetry 2023, 15(1), 240; https://doi.org/10.3390/sym15010240 - 16 Jan 2023

Cited by 2 | Viewed by 1598

Abstract

In this study, we propose the concept of left extension of a hyperideal by generalizing the concept of k-hyperideals in ordered semihyperrings with symmetrical hyper-operation ⊕. By using the notion of extension of a k-hyperideal, we prove some results in ordered [...] Read more.

In this study, we propose the concept of left extension of a hyperideal by generalizing the concept of k-hyperideals in ordered semihyperrings with symmetrical hyper-operation ⊕. By using the notion of extension of a k-hyperideal, we prove some results in ordered semihyperrings. The results of this paper can be viewed as a generalization for k-ideals of semirings. Full article

12 pages, 265 KB

Open AccessArticle

A Study on Special Kinds of Derivations in Ordered Hyperrings

by Yongsheng Rao, Saeed Kosari, Aysha Khan and Nategh Abbasizadeh

Symmetry 2022, 14(10), 2205; https://doi.org/10.3390/sym14102205 - 19 Oct 2022

Cited by 4 | Viewed by 1663

Abstract

In this study, we concentrate on an important class of ordered hyperstructures with symmetrical hyperoperations, which are called ordered Krasner hyperrings, and discuss strong derivations and homo-derivations. Additionally, we apply our results on nonzero proper hyperideals to the study of derivations of prime [...] Read more.

In this study, we concentrate on an important class of ordered hyperstructures with symmetrical hyperoperations, which are called ordered Krasner hyperrings, and discuss strong derivations and homo-derivations. Additionally, we apply our results on nonzero proper hyperideals to the study of derivations of prime ordered hyperrings. This work is a pioneer in studies on structures such as hyperideals and homomorphisms of an ordered hyperring with the help of derivation notation. Finally, we prove some results on 2-torsion-free prime ordered hyperrings by using derivations. We show that if d is a derivation of 2-torsion-free prime hyperring R and the commutator set

[l, d (q)]

is equal to zero for all q in R, then

l \in Z (R)

. Moreover, we prove that if the commutator set

(d (l), q)

is equal to zero for all l in R, then

(d (R), q) = 0

. Full article

(This article belongs to the Section Mathematics)

11 pages, 286 KB

Open AccessArticle

Multipolar Fuzzy Hyperideals in Ordered Semihypergroups

by Osman Kazancı, Sarka Hoskova-Mayerova and Bijan Davvaz

Mathematics 2022, 10(19), 3424; https://doi.org/10.3390/math10193424 - 21 Sep 2022

Cited by 4 | Viewed by 1293

Abstract

An multi-polar fuzzy set is a robust mathematical model to examine multipolar, multiattribute, and multi-index data. The multi-polar fuzzy sets was created as a useful mechanism to portray uncertainty in multiattribute decision making. In this article, we consider the theoretical applications of multi-polar [...] Read more.

An multi-polar fuzzy set is a robust mathematical model to examine multipolar, multiattribute, and multi-index data. The multi-polar fuzzy sets was created as a useful mechanism to portray uncertainty in multiattribute decision making. In this article, we consider the theoretical applications of multi-polar fuzzy sets. We present the notion of multi-polar fuzzy sets in ordered semihypergroups and define multi-polar fuzzy hyperideals (bi-hyperideals, quasi hyperideals) in an ordered semihypergroup. Relations between multi-polar fuzzy hyperideals, multi-polar fuzzy bi-hyperideals and multi-polar fuzzy quasi hyperideals are discussed. Full article

10 pages, 251 KB

Open AccessArticle

An Investigation on Weak Concepts in Ordered Hyperstructures

by Yongsheng Rao, Jietong Zhao, Aysha Khan, Maryam Akhoundi and Saber Omidi

Symmetry 2021, 13(12), 2300; https://doi.org/10.3390/sym13122300 - 2 Dec 2021

Cited by 1 | Viewed by 1567

Abstract

The class of weak pseudoorders and left weak interior hyperideals in ordered hyperstructures is a generalization of pseudoorders and interior hyperideals. In this work, we study the concept of weak pseudoorders and left weak interior hyperideals in ordered hyperstructures and explore some results [...] Read more.

The class of weak pseudoorders and left weak interior hyperideals in ordered hyperstructures is a generalization of pseudoorders and interior hyperideals. In this work, we study the concept of weak pseudoorders and left weak interior hyperideals in ordered hyperstructures and explore some results concerning the new defined concepts for ordered hyperrings and ordered Γ-semihypergroups. In this regards, we intend to concentrate our efforts on the relationship between the left weak interior hyperideal and interior hyperideal of an ordered hyperstructure. A characterization of a regular ordered hyperstructure via a left weak interior hyperideal is given. Finally, we characterize the notion of left weak interior simple ordered hyperstructures in terms of left weak interior hyperideals. Full article

21 pages, 314 KB

Open AccessArticle

Regularities in Ordered n-Ary Semihypergroups

by Jukkrit Daengsaen and Sorasak Leeratanavalee

Mathematics 2021, 9(16), 1857; https://doi.org/10.3390/math9161857 - 5 Aug 2021

Cited by 6 | Viewed by 2090

Abstract

This paper deals with a class of hyperstructures called ordered n-ary semihypergroups which are studied by means of j-hyperideals for all positive integers

1 \leq j \leq n

and

n \geq 3

. We first introduce the notion of (softly) left [...] Read more.

This paper deals with a class of hyperstructures called ordered n-ary semihypergroups which are studied by means of j-hyperideals for all positive integers

1 \leq j \leq n

and

n \geq 3

. We first introduce the notion of (softly) left regularity, (softly) right regularity, (softly) intra-regularity, complete regularity, generalized regularity of ordered n-ary semihypergroups and investigate their related properties. Several characterizations of them in terms of j-hyperideals are provided. Finally, the relationships between various classes of regularities in ordered n-ary semihypergroups are also established. Full article

(This article belongs to the Special Issue Algebraic, Analytic, and Computational Number Theory and Its Applications)

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12 pages, 284 KB

Open AccessArticle

On Some NeutroHyperstructures

by Madeleine Al-Tahan, Bijan Davvaz, Florentin Smarandache and Osman Anis

Symmetry 2021, 13(4), 535; https://doi.org/10.3390/sym13040535 - 25 Mar 2021

Cited by 18 | Viewed by 2264

Abstract

Neutrosophy, the study of neutralities, is a new branch of Philosophy that has applications in many different fields of science. Inspired by the idea of Neutrosophy, Smarandache introduced NeutroAlgebraicStructures (or NeutroAlgebras) by allowing the partiality and indeterminacy to be included in the structures’ [...] Read more.

Neutrosophy, the study of neutralities, is a new branch of Philosophy that has applications in many different fields of science. Inspired by the idea of Neutrosophy, Smarandache introduced NeutroAlgebraicStructures (or NeutroAlgebras) by allowing the partiality and indeterminacy to be included in the structures’ operations and/or axioms. The aim of this paper is to combine the concept of Neutrosophy with hyperstructures theory. In this regard, we introduce NeutroSemihypergroups as well as NeutroH_v-Semigroups and study their properties by providing several illustrative examples. Full article

(This article belongs to the Section Computer)

13 pages, 274 KB

Open AccessArticle

Regular Parameter Elements and Regular Local Hyperrings

by Hashem Bordbar and Irina Cristea

Mathematics 2021, 9(3), 243; https://doi.org/10.3390/math9030243 - 26 Jan 2021

Cited by 9 | Viewed by 2147

Abstract

Inspired by the concept of regular local rings in classical algebra, in this article we initiate the study of the regular parameter elements in a commutative local Noetherian hyperring. These elements provide a deep connection between the dimension of the hyperring and its [...] Read more.

Inspired by the concept of regular local rings in classical algebra, in this article we initiate the study of the regular parameter elements in a commutative local Noetherian hyperring. These elements provide a deep connection between the dimension of the hyperring and its primary hyperideals. Then, our study focusses on the concept of regular local hyperring R, with maximal hyperideal M, having the property that the dimension of R is equal to the dimension of the vectorial hyperspace

\frac{M}{M^{2}}

over the hyperfield

\frac{R}{M}

. Finally, using the regular local hyperrings, we determine the dimension of the hyperrings of fractions. Full article

(This article belongs to the Special Issue Hypercompositional Algebra and Applications)

17 pages, 331 KB

Open AccessArticle

On Minimal and Maximal Hyperidealsin n-ary Semihypergroups

by Jukkrit Daengsaen, Sorasak Leeratanavalee and Bijan Davvaz

Mathematics 2020, 8(10), 1656; https://doi.org/10.3390/math8101656 - 25 Sep 2020

Cited by 8 | Viewed by 1996

Abstract

The concept of j-hyperideals, for all positive integers

1 \leq j \leq n

and

n \geq 2

, in n-ary semihypergroups, is a generalization of the concept of left, lateral and right hyperideals in ternary semihypergroups. In this paper, we first [...] Read more.

The concept of j-hyperideals, for all positive integers

1 \leq j \leq n

and

n \geq 2

, in n-ary semihypergroups, is a generalization of the concept of left, lateral and right hyperideals in ternary semihypergroups. In this paper, we first introduce the concept of j-(0-)simple n-ary semihypergroups and discuss their related properties through terms of j-hyperideals. Furthermore, we characterize the minimality and maximality of j-hyperideals in n-ary semihypergroups and establish the relationships between the (0-)minimal, maximal j-hyperideals and the j-(0-)simple n-ary semihypergroups. Our main results are to extend and generalize the results on semihypergroups and ternary semihypergroups. Moreover, a related question raised by Petchkaew and Chinram is solved. Full article

(This article belongs to the Special Issue Algebra and Discrete Mathematics 2020)

15 pages, 327 KB

Open AccessArticle

Superring of Polynomials over a Hyperring

by Reza Ameri, Mansour Eyvazi and Sarka Hoskova-Mayerova

Mathematics 2019, 7(10), 902; https://doi.org/10.3390/math7100902 - 26 Sep 2019

Cited by 23 | Viewed by 3808

Abstract

A Krasner hyperring (for short, a hyperring) is a generalization of a ring such that the addition is multivalued and the multiplication is as usual single valued and satisfies the usual ring properties. One of the important subjects in the theory of hyperrings [...] Read more.

A Krasner hyperring (for short, a hyperring) is a generalization of a ring such that the addition is multivalued and the multiplication is as usual single valued and satisfies the usual ring properties. One of the important subjects in the theory of hyperrings is the study of polynomials over a hyperring. Recently, polynomials over hyperrings have been studied by Davvaz and Musavi, and they proved that polynomials over a hyperring constitute an additive-multiplicative hyperring that is a hyperstructure in which both addition and multiplication are multivalued and multiplication is distributive with respect to the addition. In this paper, we first show that the polynomials over a hyperring is not an additive-multiplicative hyperring, since the multiplication is not distributive with respect to addition; then, we study hyperideals of polynomials, such as prime and maximal hyperideals and prove that every principal hyperideal generated by an irreducible polynomial is maximal and Hilbert’s basis theorem holds for polynomials over a hyperring. Full article

10 pages, 265 KB

Open AccessArticle

Breakable Semihypergroups

by Dariush Heidari and Irina Cristea

Symmetry 2019, 11(1), 100; https://doi.org/10.3390/sym11010100 - 16 Jan 2019

Cited by 11 | Viewed by 3951

Abstract

In this paper, we introduce and characterize the breakable semihypergroups, a natural generalization of breakable semigroups, defined by a simple property: every nonempty subset of them is a subsemihypergroup. Then, we present and discuss on an extended version of Rédei’s theorem for semi-symmetric [...] Read more.

In this paper, we introduce and characterize the breakable semihypergroups, a natural generalization of breakable semigroups, defined by a simple property: every nonempty subset of them is a subsemihypergroup. Then, we present and discuss on an extended version of Rédei’s theorem for semi-symmetric breakable semihypergroups, proposing a different proof that improves also the theorem in the classical case of breakable semigroups. Full article

(This article belongs to the Special Issue Symmetry in Classical and Fuzzy Algebraic Hypercompositional Structures)

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