IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v11y2023i6p1289-d1090449.html
   My bibliography  Save this article

On the Borderline of Fields and Hyperfields

Author

Listed:
  • Christos G. Massouros

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

  • Gerasimos G. Massouros

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

Abstract

The hyperfield came into being due to a mathematical necessity that appeared during the study of the valuation theory of the fields by M. Krasner, who also defined the hyperring, which is related to the hyperfield in the same way as the ring is related to the field. The fields and the hyperfields, as well as the rings and the hyperrings, border on each other, and it is natural that problems and open questions arise in their boundary areas. This paper presents such occasions, and more specifically, it introduces a new class of non-finite hyperfields and hyperrings that is not isomorphic to the existing ones; it also classifies finite hyperfields as quotient hyperfields or non-quotient hyperfields, and it gives answers to the question that was raised from the isomorphic problems of the hyperfields: when can the subtraction of a field F ’s multiplicative subgroup G from itself generate F ? Furthermore, it presents a construction of a new class of hyperfields, and with regard to the problem of the isomorphism of its members to the quotient hyperfields, it raises a new question in field theory: when can the subtraction of a field F ’s multiplicative subgroup G from itself give all the elements of the field F , except the ones of its multiplicative subgroup G ?

Suggested Citation

  • Christos G. Massouros & Gerasimos G. Massouros, 2023. "On the Borderline of Fields and Hyperfields," Mathematics, MDPI, vol. 11(6), pages 1-35, March.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:6:p:1289-:d:1090449
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/11/6/1289/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/11/6/1289/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Hashem Bordbar & Irina Cristea, 2021. "Regular Parameter Elements and Regular Local Hyperrings," Mathematics, MDPI, vol. 9(3), pages 1-13, January.
    2. Irina Cristea & Milica Kankaraš, 2021. "The Reducibility Concept in General Hyperrings," Mathematics, MDPI, vol. 9(17), pages 1-14, August.
    3. Vahid Vahedi & Morteza Jafarpour & Sarka Hoskova-Mayerova & Hossein Aghabozorgi & Violeta Leoreanu-Fotea & Svajone Bekesiene, 2020. "Derived Hyperstructures from Hyperconics," Mathematics, MDPI, vol. 8(3), pages 1-15, March.
    4. Christos Massouros & Gerasimos Massouros, 2021. "An Overview of the Foundations of the Hypergroup Theory," Mathematics, MDPI, vol. 9(9), pages 1-41, April.
    5. Reza Ameri & Mansour Eyvazi & Sarka Hoskova-Mayerova, 2019. "Superring of Polynomials over a Hyperring," Mathematics, MDPI, vol. 7(10), pages 1-15, September.
    6. Gerasimos Massouros & Christos Massouros, 2020. "Hypercompositional Algebra, Computer Science and Geometry," Mathematics, MDPI, vol. 8(8), pages 1-33, August.
    7. Anastase Nakassis, 1988. "Recent results in hyperring and hyperfield theory," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 11, pages 1-12, January.
    Full references (including those not matched with items on IDEAS)

    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 G. Massouros & Christos G. Massouros, 2022. "State Machines and Hypergroups," Mathematics, MDPI, vol. 10(14), pages 1-25, July.
    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. Mario De Salvo & Dario Fasino & Domenico Freni & Giovanni Lo Faro, 2022. "Commutativity and Completeness Degrees of Weakly Complete Hypergroups," Mathematics, MDPI, vol. 10(6), pages 1-17, March.
    4. Hashem Bordbar & Irina Cristea, 2021. "Regular Parameter Elements and Regular Local Hyperrings," Mathematics, MDPI, vol. 9(3), pages 1-13, January.
    5. Ergül Türkmen & Burcu Nişancı Türkmen & Öznur Kulak, 2023. "Spectrum of Zariski Topology in Multiplication Krasner Hypermodules," Mathematics, MDPI, vol. 11(7), pages 1-10, April.
    6. Gerasimos Massouros & Christos Massouros, 2020. "Hypercompositional Algebra, Computer Science and Geometry," Mathematics, MDPI, vol. 8(8), pages 1-33, August.
    7. 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.
    8. Christos Massouros & Gerasimos Massouros, 2021. "An Overview of the Foundations of the Hypergroup Theory," Mathematics, MDPI, vol. 9(9), pages 1-41, April.

    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:11:y:2023:i:6:p:1289-:d:1090449. 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.