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An Overview of the Foundations of the Hypergroup Theory

Author

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  • Christos Massouros

    (Core Department, Euripus Campus, National and Kapodistrian University of Athens, GR 34400 Euboia, Greece)

  • Gerasimos Massouros

    (School of Social Sciences, Hellenic Open University, Aristotelous 18, GR 26335 Patra, Greece)

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 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.

Suggested Citation

  • Christos Massouros & Gerasimos Massouros, 2021. "An Overview of the Foundations of the Hypergroup Theory," Mathematics, MDPI, vol. 9(9), pages 1-41, April.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:9:p:1014-:d:546818
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    References listed on IDEAS

    as
    1. Ameri, R. & Zahedi, M.M., 1997. "Hypergroup and join space induced by a fuzzy subset," Pure Mathematics and Applications, Department of Mathematics, Corvinus University of Budapest, vol. 8(2-4), pages 155-168.
    2. Leoreanu, V., 2000. "Direct limit and inverse limit of join spaces associated with fuzzy sets," Pure Mathematics and Applications, Department of Mathematics, Corvinus University of Budapest, vol. 11(3), pages 509-516.
    3. Anastase Nakassis, 1988. "Recent results in hyperring and hyperfield theory," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 11, pages 1-12, January.
    4. Milica Kankaras & Irina Cristea, 2020. "Fuzzy Reduced Hypergroups," Mathematics, MDPI, vol. 8(2), pages 1-12, February.
    5. Mario De Salvo & Dario Fasino & Domenico Freni & Giovanni Lo Faro, 2021. "1-Hypergroups of Small Sizes," Mathematics, MDPI, vol. 9(2), pages 1-17, January.
    6. Corsini, P. & Tofan, I., 1997. "On fuzzy hypergroups," Pure Mathematics and Applications, Department of Mathematics, Corvinus University of Budapest, vol. 8(1), pages 29-37.
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    Citations

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    Cited by:

    1. Gerasimos G. Massouros & Christos G. Massouros, 2022. "State Machines and Hypergroups," Mathematics, MDPI, vol. 10(14), pages 1-25, July.
    2. Christos G. Massouros & Gerasimos G. Massouros, 2023. "On the Borderline of Fields and Hyperfields," Mathematics, MDPI, vol. 11(6), pages 1-35, March.
    3. Štěpán Křehlík & Michal Novák & Jana Vyroubalová, 2021. "From Automata to Multiautomata via Theory of Hypercompositional Structures," Mathematics, MDPI, vol. 10(1), pages 1-16, December.
    4. Dawid Edmund Kędzierski & Alessandro Linzi & Hanna Stojałowska, 2023. "Characteristic, C-Characteristic and Positive Cones in Hyperfields," Mathematics, MDPI, vol. 11(3), pages 1-20, February.

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