Isomorphism theorems in fortified transposition hypergroups

Massouros, Christos G.

doi:10.1063/1.4825940

AIP Conference Proceedings

2013

DOI: 10.1063/1.4825940

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Isomorphism theorems in fortified transposition hypergroups

Christos G. Massouros¹

Abstract: The isomorphism theorems of both groups as well as hypergroups are a significant tool for the study of these structures. C. G. Massouros proved the isomorphism theorems for quasicanonical hypergroups (or polygroups) and afterwards J. Jantosciak generalised them for transposition hypergroups using quotients modulo closed subhypergroups. But, there also exist transposition hypergroups which do not have proper closed subhypergroups. Such are the fortified transposition hypergroups of attractive elements. The isom… Show more

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Cited by 2 publications

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Hypercompositional Algebra, Computer Science and Geometry

Massouros

2020

Mathematics

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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.

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Hypercompositional Algebra, Computer Science and Geometry

Massouros

2020

Mathematics

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An Overview of the Foundations of the Hypergroup Theory

Massouros

2021

Mathematics

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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.

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Isomorphism theorems in fortified transposition hypergroups

Cited by 2 publications

References 5 publications

Hypercompositional Algebra, Computer Science and Geometry

Hypercompositional Algebra, Computer Science and Geometry

An Overview of the Foundations of the Hypergroup Theory

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