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21 pages, 775 KB

Open AccessArticle

Computation in Algebraic Hyperstructures

by Yuming Feng, Enrico Talotti and Violeta Leoreanu-Fotea

Computation 2025, 13(11), 261; https://doi.org/10.3390/computation13110261 - 4 Nov 2025

Cited by 2 | Viewed by 972

The concept of the relation

β

plays a central role in the study of hypercompositional structures. In this paper, we extend the definition of

β

to the general framework of hypergroupoids and develop an algorithm to compute its elements and its transitive closure [...] Read more.

The concept of the relation

β

plays a central role in the study of hypercompositional structures. In this paper, we extend the definition of

β

to the general framework of hypergroupoids and develop an algorithm to compute its elements and its transitive closure

β^{★}

. We then apply this algorithm to determine the

β

-class of partial identities in a hypergroup, a process equivalent to computing the heart of the given algebraic structure. Furthermore, we propose a more general algorithm that is also applicable to the case of

H_{v}

-groups. By extracting the quotient set with respect to

β^{★}

and endowing it with an appropriate group structure, we obtain the so-called fundamental group. The identity of this fundamental group can then be computed directly, yielding the heart of the structure. Full article

(This article belongs to the Section Computational Engineering)

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21 pages, 1160 KB

Open AccessArticle

Generalized Fuzzy Rough Approximations on Hypergroups

by Canan Akın, Dilek Bayrak Delice and Sultan Yamak

Mathematics 2024, 12(16), 2445; https://doi.org/10.3390/math12162445 - 6 Aug 2024

Cited by 2 | Viewed by 1184

Abstract

In this paper, we define the fuzzy set-valued homomorphisms of the canonical hypergroups as a generalization of fuzzy congruences and investigate some of their features. This structure is an extension of the definition of set-valued homomorphism defined for groups to hypergroups. With this [...] Read more.

In this paper, we define the fuzzy set-valued homomorphisms of the canonical hypergroups as a generalization of fuzzy congruences and investigate some of their features. This structure is an extension of the definition of set-valued homomorphism defined for groups to hypergroups. With this extension, it has become possible to study generalized fuzzy rough approximations in hyperalgebraic structures such as semihypergroups, polygroups, hyperrings, hypermodules, etc. This paper presents the generalized fuzzy rough approximations based on two-universe

(I, T)

-fuzzy model on canonical hypergroups. Full article

(This article belongs to the Section D2: Operations Research and Fuzzy Decision Making)

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12 pages, 297 KB

Open AccessArticle

On Cyclic LA-Hypergroups

by Shehzadi Salma Kanwal, Naveed Yaqoob, Nabilah Abughazalah and Muhammad Gulistan

Symmetry 2023, 15(9), 1668; https://doi.org/10.3390/sym15091668 - 30 Aug 2023

Cited by 1 | Viewed by 1483

Abstract

Symmetries in the context of hypergroups and their generalizations are closely related to the algebraic structures and transformations that preserve certain properties of hypergroup operations. Symmetric LA-hypergroups are indeed commutative hypergroups. This paper considers a category of cyclic hyperstructures called the cyclic LA-semihypergroup [...] Read more.

Symmetries in the context of hypergroups and their generalizations are closely related to the algebraic structures and transformations that preserve certain properties of hypergroup operations. Symmetric LA-hypergroups are indeed commutative hypergroups. This paper considers a category of cyclic hyperstructures called the cyclic LA-semihypergroup that is a conception of LA-semihypergroups and cyclic hypergroups. We inaugurate the idea of cyclic LA-hypergroups. The interconnected notions of single-power cyclic LA-hypergroups, non-single power cyclic LA-hypergroups and some of their properties are explored. Full article

14 pages, 4270 KB

Open AccessArticle

Semihypergroup-Based Graph for Modeling International Spread of COVID-n in Social Systems

by Narjes Firouzkouhi, Reza Ameri, Abbas Amini and Hashem Bordbar

Mathematics 2022, 10(23), 4405; https://doi.org/10.3390/math10234405 - 22 Nov 2022

Cited by 7 | Viewed by 2079

Abstract

Graph theoretic techniques have been widely applied to model many types of links in social systems. Also, algebraic hypercompositional structure theory has demonstrated its systematic application in some problems. Influenced by these mathematical notions, a novel semihypergroup-based graph (SBG) of [...] Read more.

Graph theoretic techniques have been widely applied to model many types of links in social systems. Also, algebraic hypercompositional structure theory has demonstrated its systematic application in some problems. Influenced by these mathematical notions, a novel semihypergroup-based graph (SBG) of

G = ⟨H, E⟩

is constructed through the fundamental relation

γ_{n}

on

H,

where semihypergroup H is appointed as the set of vertices and E is addressed as the set of edges on SBG. Indeed, two arbitrary vertices x and y are adjacent if

x γ_{n} y .

The connectivity of graph G is characterized by

x γ^{*} y,

whereby the connected components SBG of G would be exactly the elements of the fundamental group

H / γ^{*} .

Based on SBG, some fundamental characteristics of the graph such as complete, regular, Eulerian, isomorphism, and Cartesian products are discussed along with illustrative examples to clarify the relevance between semihypergroup H and its corresponding graph. Furthermore, the notions of geometric space, block, polygonal, and connected components are introduced in terms of the developed SBG. To formulate the links among individuals/countries in the wake of the COVID (coronavirus disease) pandemic, a theoretical SBG methodology is presented to analyze and simplify such social systems. Finally, the developed SBG is used to model the trend diffusion of the viral disease COVID-n in social systems (i.e., countries and individuals). Full article

(This article belongs to the Special Issue Algebraic Structures and Graph Theory)

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21 pages, 337 KB

Open AccessArticle

Representations, Translations and Reductions for Ternary Semihypergroups

by Anak Nongmanee and Sorasak Leeratanavalee

Axioms 2022, 11(11), 626; https://doi.org/10.3390/axioms11110626 - 8 Nov 2022

Cited by 4 | Viewed by 2000

Abstract

The concept of ternary semihypergroups can be considered as a natural generalization of arbitrary ternary semigroups. In fact, each ternary semigroup can be constructed to a ternary semihypergroup. In this article, we investigate some interesting algebraic properties of ternary semihypergroups induced by semihypergroups. [...] Read more.

The concept of ternary semihypergroups can be considered as a natural generalization of arbitrary ternary semigroups. In fact, each ternary semigroup can be constructed to a ternary semihypergroup. In this article, we investigate some interesting algebraic properties of ternary semihypergroups induced by semihypergroups. Then, we extend the well-known result on group theory and semigroup theory, the so-called Cayley’s theorem, to study on ternary semihypergroups. This leads us to construct the ternary semihypergroups of all multivalued full binary functions. In particular, we investigate that each element of a ternary semihypergroup induced by a semihypergroup can be represented by a multivalued full binary function. Moreover, we introduce the concept of translations for ternary semihypergroups which can be considered as a generalization of translations on ternary semigrgoups. Then, we construct ternary semihypergroups of all multivalued full functions and ternary semihypergroups via translations. So, some interesting algebraic properties are investigated. At the last section, we discover that there are ternary semihypergroups satisfying some significant conditions which can be reduced to semihypergroups. Furthermore, ternary semihypergroups with another one condition can be reduced to idempotent semihypergroups. Full article

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

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 1528

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

21 pages, 343 KB

Open AccessArticle

QM-BZ-Algebras and Quasi-Hyper BZ-Algebras

by Yudan Du and Xiaohong Zhang

Axioms 2022, 11(3), 93; https://doi.org/10.3390/axioms11030093 - 24 Feb 2022

Cited by 8 | Viewed by 3170

Abstract

B Z

-algebra, as the common generalization of

B C I

-algebra and

B C C

-algebra, is a kind of important logic algebra. Herein, the new concepts of QM-

B Z

-algebra and quasi-hyper

B Z

-algebra are proposed and their structures [...] Read more.

B Z

-algebra, as the common generalization of

B C I

-algebra and

B C C

-algebra, is a kind of important logic algebra. Herein, the new concepts of QM-

B Z

-algebra and quasi-hyper

B Z

-algebra are proposed and their structures and constructions are studied. First, the definition of QM-

B Z

-algebra is presented, and the structure of QM-

B Z

-algebra is obtained: Each QM-

B Z

-algebra is KG-union of quasi-alter

B C K

-algebra and anti-grouped

B Z

-algebra. Second, the new concepts of generalized quasi-left alter (hyper)

B Z

-algebras and QM-hyper

B Z

-algebra are introduced, and some characterizations of them are investigated. Third, the definition of quasi-hyper

B Z

-algebra is proposed, and the relationships among

B Z

-algebra, hyper

B Z

-algebra, quasi-hyper

B C I

-algebra, and quasi-hyper

B Z

-algebra are discussed. Finally, several special classes of quasi-hyper

B Z

-algebras are studied in depth and the following important results are proved: (1) an anti-grouped quasi-hyper

B Z

-algebra is an anti-grouped

B Z

-algebra; (2) every generalized anti-grouped quasi-hyper

B Z

-algebra corresponds to a semihypergroup. Full article

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

30 pages, 4869 KB

Open AccessArticle

On Cyclic Associative Semihypergroups and Neutrosophic Extended Triplet Cyclic Associative Semihypergroups

by Minghao Hu and Xiaohong Zhang

Mathematics 2022, 10(4), 535; https://doi.org/10.3390/math10040535 - 9 Feb 2022

Cited by 9 | Viewed by 2483

Abstract

This paper introduces a new concept called cyclic associative semihypergroup (CA-semihypergroup). The relationships among CA-semihypergroups, Semihypergroups and LA-semihypergroups are studied through some interesting examples. The relationships among various NET-CA-semihypergroups are also studied. The main properties of strong pure neutrosophic extended triplet CA-semihypergroups (SP-NET-CA-semihypergroups) [...] Read more.

This paper introduces a new concept called cyclic associative semihypergroup (CA-semihypergroup). The relationships among CA-semihypergroups, Semihypergroups and LA-semihypergroups are studied through some interesting examples. The relationships among various NET-CA-semihypergroups are also studied. The main properties of strong pure neutrosophic extended triplet CA-semihypergroups (SP-NET-CA-semihypergroups) are obtained. In particular, the algorithm of a generated CA-semihypergroup of order tm+n by two known CA-semihypergroups of order m and n is proven, and a CA-semihypergroup of order 19 is obtained by using a Python program. Moreover, it is proven that five different definitions, which can all be used as the definition of SP-NET-CA-Semihypergroup, are equivalent. Full article

(This article belongs to the Special Issue Algebra and Discrete Mathematics 2021)

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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 1783

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 8 | Viewed by 2421

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 2592

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 A: Computer Science)

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 10 | Viewed by 2250

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)

7 pages, 228 KB

Open AccessArticle

On Factorizable Semihypergroups

by Dariush Heidari and Irina Cristea

Mathematics 2020, 8(7), 1064; https://doi.org/10.3390/math8071064 - 1 Jul 2020

Cited by 7 | Viewed by 2164

Abstract

In this paper, we define and study the concept of the factorizable semihypergroup, i.e., a semihypergroup that can be written as a hyperproduct of two proper sub-semihypergroups. We consider some classes of semihypergroups such as regular semihypergroups, hypergroups, regular hypergroups, and polygroups and [...] Read more.

In this paper, we define and study the concept of the factorizable semihypergroup, i.e., a semihypergroup that can be written as a hyperproduct of two proper sub-semihypergroups. We consider some classes of semihypergroups such as regular semihypergroups, hypergroups, regular hypergroups, and polygroups and investigate their factorization property. Full article

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

14 pages, 306 KB

Open AccessArticle

On Hypergroups with a β-Class of Finite Height

by Mario De Salvo, Dario Fasino, Domenico Freni and Giovanni Lo Faro

Symmetry 2020, 12(1), 168; https://doi.org/10.3390/sym12010168 - 15 Jan 2020

Cited by 8 | Viewed by 2642

Abstract

In every hypergroup, the equivalence classes modulo the fundamental relation

β

are the union of hyperproducts of element pairs. Making use of this property, we introduce the notion of height of a

β

-class and we analyze properties of hypergroups where the height [...] Read more.

In every hypergroup, the equivalence classes modulo the fundamental relation

β

are the union of hyperproducts of element pairs. Making use of this property, we introduce the notion of height of a

β

-class and we analyze properties of hypergroups where the height of a

β

-class coincides with its cardinality. As a consequence, we obtain a new characterization of 1-hypergroups. Moreover, we define a hierarchy of classes of hypergroups where at least one

β

-class has height 1 or cardinality 1, and we enumerate the elements in each class when the size of the hypergroups is

n \leq 4

, apart from isomorphisms. Full article

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

22 pages, 1911 KB

Open AccessArticle

On Neutrosophic Extended Triplet LA-hypergroups and Strong Pure LA-semihypergroups

by Minghao Hu, Florentin Smarandache and Xiaohong Zhang

Symmetry 2020, 12(1), 163; https://doi.org/10.3390/sym12010163 - 14 Jan 2020

Cited by 5 | Viewed by 2762

Abstract

We introduce the notions of neutrosophic extended triplet LA-semihypergroup, neutrosophic extended triplet LA-hypergroup, which can reflect some symmetry of hyperoperation and discuss the relationships among them and regular LA-semihypergroups, LA-hypergroups, regular LA-hypergroups. In particular, we introduce the notion of strong pure neutrosophic extended [...] Read more.

We introduce the notions of neutrosophic extended triplet LA-semihypergroup, neutrosophic extended triplet LA-hypergroup, which can reflect some symmetry of hyperoperation and discuss the relationships among them and regular LA-semihypergroups, LA-hypergroups, regular LA-hypergroups. In particular, we introduce the notion of strong pure neutrosophic extended triplet LA-semihypergroup, get some special properties of it and prove the construction theorem about it under the condition of asymmetry. The examples in this paper are all from Python programs. Full article

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