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3 pages, 183 KiB

Open AccessEditorial

Preface to the Special Issue on “Hypergroup Theory and Algebrization of Incidence Structures”

by Dario Fasino and Domenico Freni

Mathematics 2023, 11(15), 3424; https://doi.org/10.3390/math11153424 - 7 Aug 2023

Viewed by 758

Abstract

This work contains the accepted papers of a Special Issue of the MDPI journal Mathematics entitled “Hypergroup Theory and Algebrization of Incidence Structure” [...] Full article

(This article belongs to the Special Issue Hypergroup Theory and Algebrization of Incidence Structures)

Research

Jump to: Editorial

35 pages, 11250 KiB

Open AccessArticle

On the Borderline of Fields and Hyperfields

by Christos G. Massouros and Gerasimos G. Massouros

Mathematics 2023, 11(6), 1289; https://doi.org/10.3390/math11061289 - 7 Mar 2023

Cited by 6 | Viewed by 1363

Abstract

The hyperfield came into being due to a mathematical necessity that appeared during the study of the valuation theory of the fields by M. Krasner, who also defined the hyperring, which is related to the hyperfield in the same way as the ring [...] Read more.

The hyperfield came into being due to a mathematical necessity that appeared during the study of the valuation theory of the fields by M. Krasner, who also defined the hyperring, which is related to the hyperfield in the same way as the ring is related to the field. The fields and the hyperfields, as well as the rings and the hyperrings, border on each other, and it is natural that problems and open questions arise in their boundary areas. This paper presents such occasions, and more specifically, it introduces a new class of non-finite hyperfields and hyperrings that is not isomorphic to the existing ones; it also classifies finite hyperfields as quotient hyperfields or non-quotient hyperfields, and it gives answers to the question that was raised from the isomorphic problems of the hyperfields: when can the subtraction of a field F’s multiplicative subgroup G from itself generate F? Furthermore, it presents a construction of a new class of hyperfields, and with regard to the problem of the isomorphism of its members to the quotient hyperfields, it raises a new question in field theory: when can the subtraction of a field F’s multiplicative subgroup G from itself give all the elements of the field F, except the ones of its multiplicative subgroup G? Full article

(This article belongs to the Special Issue Hypergroup Theory and Algebrization of Incidence Structures)

16 pages, 315 KiB

Open AccessArticle

Algebraic Hyperstructure of Multi-Fuzzy Soft Sets Related to Polygroups

by Osman Kazancı, Sarka Hoskova-Mayerova and Bijan Davvaz

Mathematics 2022, 10(13), 2178; https://doi.org/10.3390/math10132178 - 22 Jun 2022

Cited by 7 | Viewed by 1553

Abstract

The combination of two elements in a group structure is an element, while, in a hypergroup, the combination of two elements is a non-empty set. The use of hypergroups appears mainly in certain subclasses. For instance, polygroups, which are a special subcategory of [...] Read more.

The combination of two elements in a group structure is an element, while, in a hypergroup, the combination of two elements is a non-empty set. The use of hypergroups appears mainly in certain subclasses. For instance, polygroups, which are a special subcategory of hypergroups, are used in many branches of mathematics and basic sciences. On the other hand, in a multi-fuzzy set, an element of a universal set may occur more than once with possibly the same or different membership values. A soft set over a universal set is a mapping from parameters to the family of subsets of the universal set. If we substitute the set of all fuzzy subsets of the universal set instead of crisp subsets, then we obtain fuzzy soft sets. Similarly, multi-fuzzy soft sets can be obtained. In this paper, we combine the multi-fuzzy soft set and polygroup structure, from which we obtain a new soft structure called the multi-fuzzy soft polygroup. We analyze the relation between multi-fuzzy soft sets and polygroups. Some algebraic properties of fuzzy soft polygroups and soft polygroups are extended to multi-fuzzy soft polygroups. Some new operations on a multi-fuzzy soft set are defined. In addition to this, we investigate normal multi-fuzzy soft polygroups and present some of their algebraic properties. Full article

(This article belongs to the Special Issue Hypergroup Theory and Algebrization of Incidence Structures)

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9 pages, 266 KiB

Open AccessFeature PaperArticle

A Combinatorial Characterization of H(4, q²)

by Stefano Innamorati and Fulvio Zuanni

Mathematics 2022, 10(10), 1707; https://doi.org/10.3390/math10101707 - 16 May 2022

Cited by 1 | Viewed by 1354

Abstract

In this paper, we remove the solid incidence assumption in a characterization of

H (4, q^{2})

by J. Schillewaert and J. A. Thasby proving that Hermitian plane incidence numbers imply Hermitian solid incidence numbers, except for a few possible small cases. Full article

(This article belongs to the Special Issue Hypergroup Theory and Algebrization of Incidence Structures)

13 pages, 291 KiB

Open AccessArticle

Construction of an Infinite Cyclic Group Formed by Artificial Differential Neurons

by Jan Chvalina, Bedřich Smetana and Jana Vyroubalová

Mathematics 2022, 10(9), 1571; https://doi.org/10.3390/math10091571 - 6 May 2022

Cited by 1 | Viewed by 1231

Abstract

Infinite cyclic groups created by various objects belong to the class to the class basic algebraic structures. In this paper, we construct the infinite cyclic group of differential neurons which are modifications of artificial neurons in analogy to linear ordinary differential operators of [...] Read more.

Infinite cyclic groups created by various objects belong to the class to the class basic algebraic structures. In this paper, we construct the infinite cyclic group of differential neurons which are modifications of artificial neurons in analogy to linear ordinary differential operators of the n-th order. We also describe some of their basic properties. Full article

(This article belongs to the Special Issue Hypergroup Theory and Algebrization of Incidence Structures)

17 pages, 353 KiB

Open AccessArticle

Commutativity and Completeness Degrees of Weakly Complete Hypergroups

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

Mathematics 2022, 10(6), 981; https://doi.org/10.3390/math10060981 - 18 Mar 2022

Cited by 5 | Viewed by 1604

Abstract

We introduce a family of hypergroups, called weakly complete, generalizing the construction of complete hypergroups. Starting from a given group G, our construction prescribes the

β

-classes of the hypergroups and allows some hyperproducts not to be complete parts, based on a [...] Read more.

We introduce a family of hypergroups, called weakly complete, generalizing the construction of complete hypergroups. Starting from a given group G, our construction prescribes the

β

-classes of the hypergroups and allows some hyperproducts not to be complete parts, based on a suitably defined relation over G. The commutativity degree of weakly complete hypergroups can be related to that of the underlying group. Furthermore, in analogy to the degree of commutativity, we introduce the degree of completeness of finite hypergroups and analyze this degree for weakly complete hypergroups in terms of their

β

-classes. Full article

(This article belongs to the Special Issue Hypergroup Theory and Algebrization of Incidence Structures)

7 pages, 276 KiB

Open AccessArticle

Uniform (C_k, P_k+1)-Factorizations of K_n − I When k Is Even

by Giovanni Lo Faro, Salvatore Milici and Antoinette Tripodi

Mathematics 2022, 10(6), 936; https://doi.org/10.3390/math10060936 - 15 Mar 2022

Cited by 2 | Viewed by 1549

Abstract

Let H be a connected subgraph of a graph G. An H-factor of G is a spanning subgraph of G whose components are isomorphic to H. Given a set

H

of mutually non-isomorphic graphs, a uniform

H

-factorization of G [...] Read more.

Let H be a connected subgraph of a graph G. An H-factor of G is a spanning subgraph of G whose components are isomorphic to H. Given a set

H

of mutually non-isomorphic graphs, a uniform

H

-factorization of G is a partition of the edges of G into H-factors for some

H \in H

. In this article, we give a complete solution to the existence problem for uniform

(C_{k}, P_{k + 1})

-factorizations of

K_{n} - I

in the case when k is even. Full article

(This article belongs to the Special Issue Hypergroup Theory and Algebrization of Incidence Structures)

17 pages, 374 KiB

Open AccessArticle

G-Hypergroups: Hypergroups with a Group-Isomorphic Heart

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

Mathematics 2022, 10(2), 240; https://doi.org/10.3390/math10020240 - 13 Jan 2022

Cited by 2 | Viewed by 1334

Abstract

Hypergroups can be subdivided into two large classes: those whose heart coincide with the entire hypergroup and those in which the heart is a proper sub-hypergroup. The latter class includes the family of 1-hypergroups, whose heart reduces to a singleton, and therefore is [...] Read more.

Hypergroups can be subdivided into two large classes: those whose heart coincide with the entire hypergroup and those in which the heart is a proper sub-hypergroup. The latter class includes the family of 1-hypergroups, whose heart reduces to a singleton, and therefore is the trivial group. However, very little is known about hypergroups that are neither 1-hypergroups nor belong to the first class. The goal of this work is to take a first step in classifying G-hypergroups, that is, hypergroups whose heart is a nontrivial group. We introduce their main properties, with an emphasis on G-hypergroups whose the heart is a torsion group. We analyze the main properties of the stabilizers of group actions of the heart, which play an important role in the construction of multiplicative tables of G-hypergroups. Based on these results, we characterize the G-hypergroups that are of type U on the right or cogroups on the right. Finally, we present the hyperproduct tables of all G-hypergroups of size not larger than 5, apart of isomorphisms. Full article

(This article belongs to the Special Issue Hypergroup Theory and Algebrization of Incidence Structures)

12 pages, 282 KiB

Open AccessArticle

v-Regular Ternary Menger Algebras and Left Translations of Ternary Menger Algebras

by Anak Nongmanee and Sorasak Leeratanavalee

Mathematics 2021, 9(21), 2691; https://doi.org/10.3390/math9212691 - 22 Oct 2021

Cited by 7 | Viewed by 1634

Abstract

Let n be a fixed natural number. Ternary Menger algebras of rank n, which was established by the authors, can be regarded as a suitable generalization of ternary semigroups. In this article, we introduce the notion of v-regular ternary Menger algebras [...] Read more.

Let n be a fixed natural number. Ternary Menger algebras of rank n, which was established by the authors, can be regarded as a suitable generalization of ternary semigroups. In this article, we introduce the notion of v-regular ternary Menger algebras of rank n, which can be considered as a generalization of regular ternary semigroups. Moreover, we investigate some of its interesting properties. Based on the concept of n-place functions (n-ary operations), these lead us to construct ternary Menger algebras of rank n of all full n-place functions. Finally, we study a special class of full n-place functions, the so-called left translations. In particular, we investigate a relationship between the concept of full n-place functions and left translations. Full article

(This article belongs to the Special Issue Hypergroup Theory and Algebrization of Incidence Structures)

18 pages, 796 KiB

Open AccessArticle

Taxonomy of Polar Subspaces of Multi-Qubit Symplectic Polar Spaces of Small Rank

by Metod Saniga, Henri de Boutray, Frédéric Holweck and Alain Giorgetti

Mathematics 2021, 9(18), 2272; https://doi.org/10.3390/math9182272 - 16 Sep 2021

Cited by 5 | Viewed by 4540

Abstract

We study certain physically-relevant subgeometries of binary symplectic polar spaces

W (2 N - 1, 2)

of small rank N, when the points of these spaces canonically encode N-qubit observables. Key characteristics of a subspace of such a [...] Read more.

We study certain physically-relevant subgeometries of binary symplectic polar spaces

W (2 N - 1, 2)

of small rank N, when the points of these spaces canonically encode N-qubit observables. Key characteristics of a subspace of such a space

W (2 N - 1, 2)

are: the number of its negative lines, the distribution of types of observables, the character of the geometric hyperplane the subspace shares with the distinguished (non-singular) quadric of

W (2 N - 1, 2)

and the structure of its Veldkamp space. In particular, we classify and count polar subspaces of

W (2 N - 1, 2)

whose rank is

N - 1

.

W (3, 2)

features three negative lines of the same type and its

W (1, 2)

’s are of five different types.

W (5, 2)

is endowed with 90 negative lines of two types and its

W (3, 2)

’s split into 13 types. A total of 279 out of 480

W (3, 2)

’s with three negative lines are composite, i.e., they all originate from the two-qubit

W (3, 2)

. Given a three-qubit

W (3, 2)

and any of its geometric hyperplanes, there are three other

W (3, 2)

’s possessing the same hyperplane. The same holds if a geometric hyperplane is replaced by a ‘planar’ tricentric triad. A hyperbolic quadric of

W (5, 2)

is found to host particular sets of seven

W (3, 2)

’s, each of them being uniquely tied to a Conwell heptad with respect to the quadric. There is also a particular type of

W (3, 2)

’s, a representative of which features a point each line through which is negative. Finally,

W (7, 2)

is found to possess 1908 negative lines of five types and its

W (5, 2)

’s fall into as many as 29 types. A total of 1524 out of 1560

W (5, 2)

’s with 90 negative lines originate from the three-qubit

W (5, 2)

. Remarkably, the difference in the number of negative lines for any two distinct types of four-qubit

W (5, 2)

’s is a multiple of four. Full article

(This article belongs to the Special Issue Hypergroup Theory and Algebrization of Incidence Structures)

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14 pages, 306 KiB

Open AccessArticle

The Reducibility Concept in General Hyperrings

by Irina Cristea and Milica Kankaraš

Mathematics 2021, 9(17), 2037; https://doi.org/10.3390/math9172037 - 24 Aug 2021

Cited by 5 | Viewed by 1599

Abstract

By using three equivalence relations, we characterize the behaviour of the elements in a hypercompositional structure. With respect to a hyperoperation, some elements play specific roles: their hypercomposition with all the elements of the carrier set gives the same result; they belong to [...] Read more.

By using three equivalence relations, we characterize the behaviour of the elements in a hypercompositional structure. With respect to a hyperoperation, some elements play specific roles: their hypercomposition with all the elements of the carrier set gives the same result; they belong to the same hypercomposition of elements; or they have both properties, being essentially indistinguishable. These equivalences were first defined for hypergroups, and here we extend and study them for general hyperrings—that is, structures endowed with two hyperoperations. We first present their general properties, we define the concept of reducibility, and then we focus on particular classes of hyperrings: the hyperrings of formal series, the hyperrings with P-hyperoperations, complete hyperrings, and

(H, R)

-hyperrings. Our main aim is to find conditions under which these hyperrings are reduced or not. Full article

(This article belongs to the Special Issue Hypergroup Theory and Algebrization of Incidence Structures)

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