IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v9y2021i9p1014-d546818.html
   My bibliography  Save this article

An Overview of the Foundations of the Hypergroup Theory

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/9/9/1014/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/9/9/1014/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. 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.
    2. 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.
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Christos G. Massouros & Gerasimos G. Massouros, 2023. "On the Borderline of Fields and Hyperfields," Mathematics, MDPI, vol. 11(6), pages 1-35, March.
    2. 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.
    3. 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.
    4. Gerasimos G. Massouros & Christos G. Massouros, 2022. "State Machines and Hypergroups," Mathematics, MDPI, vol. 10(14), pages 1-25, July.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Gerasimos Massouros & Christos Massouros, 2020. "Hypercompositional Algebra, Computer Science and Geometry," Mathematics, MDPI, vol. 8(8), pages 1-33, August.
    2. Irina Cristea & Milica Kankaraš, 2021. "The Reducibility Concept in General Hyperrings," Mathematics, MDPI, vol. 9(17), pages 1-14, August.
    3. Christos G. Massouros & Naveed Yaqoob, 2021. "On the Theory of Left/Right Almost Groups and Hypergroups with their Relevant Enumerations," Mathematics, MDPI, vol. 9(15), pages 1-31, August.
    4. Christos G. Massouros & Gerasimos G. Massouros, 2023. "On the Borderline of Fields and Hyperfields," Mathematics, MDPI, vol. 11(6), pages 1-35, March.
    5. Hashem Bordbar & Irina Cristea, 2021. "Regular Parameter Elements and Regular Local Hyperrings," Mathematics, MDPI, vol. 9(3), pages 1-13, January.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:9:y:2021:i:9:p:1014-:d:546818. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.