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The Reducibility Concept in General Hyperrings

Author

Listed:
  • Irina Cristea

    (Centre for Information Technologies and Applied Mathematics, University of Nova Gorica, 5000 Nova Gorica, Slovenia
    These authors contributed equally to this work.)

  • Milica Kankaraš

    (Department of Mathematics, University of Montenegro, 81000 Podgorica, Montenegro
    These authors contributed equally to this work.)

Abstract

By using three equivalence relations, we characterize the behaviour of the elements in a hypercompositional structure. With respect to a hyperoperation, some elements play specific roles: their hypercomposition with all the elements of the carrier set gives the same result; they belong to the same hypercomposition of elements; or they have both properties, being essentially indistinguishable. These equivalences were first defined for hypergroups, and here we extend and study them for general hyperrings—that is, structures endowed with two hyperoperations. We first present their general properties, we define the concept of reducibility, and then we focus on particular classes of hyperrings: the hyperrings of formal series, the hyperrings with P -hyperoperations, complete hyperrings, and ( H , R ) -hyperrings. Our main aim is to find conditions under which these hyperrings are reduced or not.

Suggested Citation

  • Irina Cristea & Milica Kankaraš, 2021. "The Reducibility Concept in General Hyperrings," Mathematics, MDPI, vol. 9(17), pages 1-14, August.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:17:p:2037-:d:621076
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    References listed on IDEAS

    as
    1. Milica Kankaras & Irina Cristea, 2020. "Fuzzy Reduced Hypergroups," Mathematics, MDPI, vol. 8(2), pages 1-12, February.
    2. 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.
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    Citations

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    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.

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