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: 30 June 2020.

Special Issue Editor

Prof. Christos G. Massouros
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. Christos G. Massouros
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All papers will be peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

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Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1200 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

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

Published Papers (3 papers)

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Research

Open AccessArticle
Derived Hyperstructures from Hyperconics
Mathematics 2020, 8(3), 429; https://doi.org/10.3390/math8030429 - 16 Mar 2020
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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Open AccessArticle
Fuzzy Reduced Hypergroups
Mathematics 2020, 8(2), 263; https://doi.org/10.3390/math8020263 - 17 Feb 2020
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)
Open AccessArticle
Fuzzy Multi-Hypergroups
Mathematics 2020, 8(2), 244; https://doi.org/10.3390/math8020244 - 14 Feb 2020
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)
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