Special Issue "Hypercompositional Algebra and Applications"

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Fuzzy Set Theory".

Deadline for manuscript submissions: closed (31 December 2020).

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A printed edition of this Special Issue is available here.

Special Issue Editor

Prof. Dr. Christos G. Massouros
E-Mail Website
Guest Editor
National and Kapodistrian University of Athens, Euripus Campus, GR34400 Euboea, Greece
Interests: hypercompositional algebra and its applications in geometry; computer theory; discrete mathematics

Special Issue Information

Dear Colleagues,

This Special Issue is about Hypercompositional Algebra, which is a recent branch of abstract algebra.

As it is known, algebra is a generalization of arithmetic, where arithmetic (from the Greek word «ἀριθμός»—arithmós—meaning «number») is the area of mathematics that deals with numbers. During the classical period, Greek mathematicians created a geometric algebra where terms were represented by sides of geometric objects, while the Alexandrian School that followed (which was founded during the Hellenistic era) changed this approach dramatically. Especially, Diophantus made the first fundamental step towards symbolic algebra, as he developed a mathematical notation in order to write and solve algebraic equations.

In the Middle Ages that followed, the stage of mathematics shifted from the Greek world to the Arabic, and in AD 830 the Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī wrote his famous treatise al-Kitāb al-mukhtaar fī isāb al-jabr wal-muqābala. After the 15th century AD, the stage of mathematics shifted to Europe, as the Islamic world was in decline and the European world was ascending. However, algebra remained essentially arithmetic with non-numerical mathematical objects until the 19th century, when the algebraic mathematical thought was changed radically because of the work of two young mathematicians: the Norwegian Niels Henrik Abel (1802–1829) and the Frenchman Evariste Galois (1811–1832), in algebraic equations. It became understood that the same processes could be applied to various objects or sets of entities other than numbers. So, abstract algebra was born.

The early years of the next (20th) century brought the end of determinism and certainty to science. This uncertainty affected algebra as well, via the work of a young French mathematician, Frederic Marty (1911–1940), who introduced an algebraic structure in which the rule of synthesizing elements gives a set of elements instead of one element only. He called this structure ‘’hypergroup’’, and he presented it during the 8th congress of Scandinavian Mathematicians, held in Stockholm in 1934. Unfortunately, Marty was killed at the age of 29, when his airplane was hit over the Baltic Sea while he was in military duty during World War II. His mathematical heritage on hypergroups is three papers only. However, other mathematicians such as H. Wall, M. Dresher, O. Ore, M. Krasner, and M. Kuntzmann started working on hypergroups shortly thereafter. Thus, hypercompositional algebra came into being as a branch of abstract algebra that deals with structures endowed with multi-valued operations. Multi-valued operations, also called ‘’hyperoperations’’ or ‘’hypercompositions’’, are laws of synthesis of the elements of a nonempty set, which associates a set of elements, instead of a single element, with every pair of elements.

Nowadays, this theory is characterized by huge diversity of character and content, and can present results in mathematics and other sciences such as physics, chemistry, biology, computer science, information technologies, social sciences, etc.

The aim of this Special Issue of the Journal Mathematics is dual: (a) to collect original research papers with new ideas and results on the contemporary research areas of hypercompositional algebra, as well as its applications to mathematics and other sciences and (b) to collect well-organized reviews on the aforementioned topics.

Prof. Dr. Christos G. Massouros
Guest Editor

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Keywords

  • Semihypergroup
  • Quasihypergroup
  • Hypergroup
  • Hyperring
  • Hyperfield
  • Hypermodule
  • Fuzzy hypercompositional structures
  • Convex sets
  • Graphs
  • Formal languages
  • Automata

Published Papers (11 papers)

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Research

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Article
On the Theory of Left/Right Almost Groups and Hypergroups with their Relevant Enumerations
Mathematics 2021, 9(15), 1828; https://doi.org/10.3390/math9151828 - 03 Aug 2021
Viewed by 445
Abstract
This paper presents the study of algebraic structures equipped with the inverted associativity axiom. Initially, the definition of the left and the right almost-groups is introduced and afterwards, the study is focused on the more general structures, which are the left and the [...] Read more.
This paper presents the study of algebraic structures equipped with the inverted associativity axiom. Initially, the definition of the left and the right almost-groups is introduced and afterwards, the study is focused on the more general structures, which are the left and the right almost-hypergroups and on their enumeration in the cases of order 2 and 3. The outcomes of these enumerations compared with the corresponding in the hypergroups reveal interesting results. Next, fundamental properties of the left and right almost-hypergroups are proved. Subsequently, the almost hypergroups are enriched with more axioms, like the transposition axiom and the weak commutativity. This creates new hypercompositional structures, such as the transposition left/right almost-hypergroups, the left/right almost commutative hypergroups, the join left/right almost hypergroups, etc. The algebraic properties of these new structures are analyzed and studied as well. Especially, the existence of neutral elements leads to the separation of their elements into attractive and non-attractive ones. If the existence of the neutral element is accompanied with the existence of symmetric elements as well, then the fortified transposition left/right almost-hypergroups and the transposition polysymmetrical left/right almost-hypergroups come into being. Full article
(This article belongs to the Special Issue Hypercompositional Algebra and Applications)
Article
Sequences of Groups, Hypergroups and Automata of Linear Ordinary Differential Operators
Mathematics 2021, 9(4), 319; https://doi.org/10.3390/math9040319 - 05 Feb 2021
Cited by 2 | Viewed by 439
Abstract
The main objective of our paper is to focus on the study of sequences (finite or countable) of groups and hypergroups of linear differential operators of decreasing orders. By using a suitable ordering or preordering of groups linear differential operators we construct hypercompositional [...] Read more.
The main objective of our paper is to focus on the study of sequences (finite or countable) of groups and hypergroups of linear differential operators of decreasing orders. By using a suitable ordering or preordering of groups linear differential operators we construct hypercompositional structures of linear differential operators. Moreover, we construct actions of groups of differential operators on rings of polynomials of one real variable including diagrams of actions–considered as special automata. Finally, we obtain sequences of hypergroups and automata. The examples, we choose to explain our theoretical results with, fall within the theory of artificial neurons and infinite cyclic groups. Full article
(This article belongs to the Special Issue Hypercompositional Algebra and Applications)
Article
Regular Parameter Elements and Regular Local Hyperrings
Mathematics 2021, 9(3), 243; https://doi.org/10.3390/math9030243 - 26 Jan 2021
Cited by 2 | Viewed by 476
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 MM2 over the hyperfield RM. 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)
Article
1-Hypergroups of Small Sizes
Mathematics 2021, 9(2), 108; https://doi.org/10.3390/math9020108 - 06 Jan 2021
Cited by 3 | Viewed by 404
Abstract
In this paper, we show a new construction of hypergroups that, under appropriate conditions, are complete hypergroups or non-complete 1-hypergroups. Furthermore, we classify the 1-hypergroups of size 5 and 6 based on the partition induced by the fundamental relation β. Many of [...] Read more.
In this paper, we show a new construction of hypergroups that, under appropriate conditions, are complete hypergroups or non-complete 1-hypergroups. Furthermore, we classify the 1-hypergroups of size 5 and 6 based on the partition induced by the fundamental relation β. Many of these hypergroups can be obtained using the aforesaid hypergroup construction. Full article
(This article belongs to the Special Issue Hypercompositional Algebra and Applications)
Article
Hypercompositional Algebra, Computer Science and Geometry
Mathematics 2020, 8(8), 1338; https://doi.org/10.3390/math8081338 - 11 Aug 2020
Cited by 7 | Viewed by 579
Abstract
The various branches of Mathematics are not separated between themselves. On the contrary, they interact and extend into each other’s sometimes seemingly different and unrelated areas and help them advance. In this sense, the Hypercompositional Algebra’s path has crossed, among others, with the [...] Read more.
The various branches of Mathematics are not separated between themselves. On the contrary, they interact and extend into each other’s sometimes seemingly different and unrelated areas and help them advance. In this sense, the Hypercompositional Algebra’s path has crossed, among others, with the paths of the theory of Formal Languages, Automata and Geometry. This paper presents the course of development from the hypergroup, as it was initially defined in 1934 by F. Marty to the hypergroups which are endowed with more axioms and allow the proof of Theorems and Propositions that generalize Kleen’s Theorem, determine the order and the grade of the states of an automaton, minimize it and describe its operation. The same hypergroups lie underneath Geometry and they produce results which give as Corollaries well known named Theorems in Geometry, like Helly’s Theorem, Kakutani’s Lemma, Stone’s Theorem, Radon’s Theorem, Caratheodory’s Theorem and Steinitz’s Theorem. This paper also highlights the close relationship between the hyperfields and the hypermodules to geometries, like projective geometries and spherical geometries. Full article
(This article belongs to the Special Issue Hypercompositional Algebra and Applications)
Article
On Factorizable Semihypergroups
Mathematics 2020, 8(7), 1064; https://doi.org/10.3390/math8071064 - 01 Jul 2020
Cited by 2 | Viewed by 538
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)
Article
n-Ary Cartesian Composition of Multiautomata with Internal Link for Autonomous Control of Lane Shifting
Mathematics 2020, 8(5), 835; https://doi.org/10.3390/math8050835 - 21 May 2020
Cited by 2 | Viewed by 830
Abstract
In this paper, which is based on a real-life motivation, we present an algebraic theory of automata and multi-automata. We combine these (multi-)automata using the products introduced by W. Dörfler, where we work with the cartesian composition and we define the internal links [...] Read more.
In this paper, which is based on a real-life motivation, we present an algebraic theory of automata and multi-automata. We combine these (multi-)automata using the products introduced by W. Dörfler, where we work with the cartesian composition and we define the internal links among multiautomata by means of the internal links’ matrix. We used the obtained product of n-ary multi-automata as a system that models and controls certain traffic situations (lane shifting) for autonomous vehicles. Full article
(This article belongs to the Special Issue Hypercompositional Algebra and Applications)
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Article
Derived Hyperstructures from Hyperconics
Mathematics 2020, 8(3), 429; https://doi.org/10.3390/math8030429 - 16 Mar 2020
Cited by 5 | Viewed by 958
Abstract
In this paper, we introduce generalized quadratic forms and hyperconics over quotient hyperfields as a generalization of the notion of conics on fields. Conic curves utilized in cryptosystems; in fact the public key cryptosystem is based on the digital signature schemes (DLP) in [...] Read more.
In this paper, we introduce generalized quadratic forms and hyperconics over quotient hyperfields as a generalization of the notion of conics on fields. Conic curves utilized in cryptosystems; in fact the public key cryptosystem is based on the digital signature schemes (DLP) in conic curve groups. We associate some hyperoperations to hyperconics and investigate their properties. At the end, a collection of canonical hypergroups connected to hyperconics is proposed. Full article
(This article belongs to the Special Issue Hypercompositional Algebra and Applications)
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Article
Fuzzy Reduced Hypergroups
Mathematics 2020, 8(2), 263; https://doi.org/10.3390/math8020263 - 17 Feb 2020
Cited by 6 | Viewed by 676
Abstract
The fuzzyfication of hypercompositional structures has developed in several directions. In this note we follow one direction and extend the classical concept of reducibility in hypergroups to the fuzzy case. In particular we define and study the fuzzy reduced hypergroups. New fundamental relations [...] Read more.
The fuzzyfication of hypercompositional structures has developed in several directions. In this note we follow one direction and extend the classical concept of reducibility in hypergroups to the fuzzy case. In particular we define and study the fuzzy reduced hypergroups. New fundamental relations are defined on a crisp hypergroup endowed with a fuzzy set, that lead to the concept of fuzzy reduced hypergroup. This is a hypergroup in which the equivalence class of any element, with respect to a determined fuzzy set, is a singleton. The most well known fuzzy set considered on a hypergroup is the grade fuzzy set, used for the study of the fuzzy grade of a hypergroup. Based on this, in the second part of the paper, we study the fuzzy reducibility of some particular classes of crisp hypergroups with respect to the grade fuzzy set. Full article
(This article belongs to the Special Issue Hypercompositional Algebra and Applications)
Article
Fuzzy Multi-Hypergroups
Mathematics 2020, 8(2), 244; https://doi.org/10.3390/math8020244 - 14 Feb 2020
Cited by 5 | Viewed by 674
Abstract
A fuzzy multiset is a generalization of a fuzzy set. This paper aims to combine the innovative notion of fuzzy multisets and hypergroups. In particular, we use fuzzy multisets to introduce the concept of fuzzy multi-hypergroups as a generalization of fuzzy hypergroups. Different [...] Read more.
A fuzzy multiset is a generalization of a fuzzy set. This paper aims to combine the innovative notion of fuzzy multisets and hypergroups. In particular, we use fuzzy multisets to introduce the concept of fuzzy multi-hypergroups as a generalization of fuzzy hypergroups. Different operations on fuzzy multi-hypergroups are defined and discussed and some results known for fuzzy hypergroups are generalized to fuzzy multi-hypergroups. Full article
(This article belongs to the Special Issue Hypercompositional Algebra and Applications)

Review

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Review
An Overview of the Foundations of the Hypergroup Theory
Mathematics 2021, 9(9), 1014; https://doi.org/10.3390/math9091014 - 30 Apr 2021
Cited by 2 | Viewed by 469
Abstract
This paper is written in the framework of the Special Issue of Mathematics entitled “Hypercompositional Algebra and Applications”, and focuses on the presentation of the essential principles of the hypergroup, which is the prominent structure of hypercompositional algebra. In the beginning, it reveals [...] Read more.
This paper is written in the framework of the Special Issue of Mathematics entitled “Hypercompositional Algebra and Applications”, and focuses on the presentation of the essential principles of the hypergroup, which is the prominent structure of hypercompositional algebra. In the beginning, it reveals the structural relation between two fundamental entities of abstract algebra, the group and the hypergroup. Next, it presents the several types of hypergroups, which derive from the enrichment of the hypergroup with additional axioms besides the ones it was initially equipped with, along with their fundamental properties. Furthermore, it analyzes and studies the various subhypergroups that can be defined in hypergroups in combination with their ability to decompose the hypergroups into cosets. The exploration of this far-reaching concept highlights the particularity of the hypergroup theory versus the abstract group theory, and demonstrates the different techniques and special tools that must be developed in order to achieve results on hypercompositional algebra. Full article
(This article belongs to the Special Issue Hypercompositional Algebra and Applications)
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