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AIMS Mathematics

2025, Volume 10, Issue 9: 21287-21421. doi: 10.3934/math.2025951
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On the borderline of fields and hyperfields, part Ⅱ – Enumeration and classification of the hyperfields of order 7

  • 1.
    Core Department, National and Kapodistrian University of Athens, GR 34400 Euboia, Greece
  • 2.
    School of Social Sciences, Hellenic Open University, GR 26335 Patra, Greece
  • Published: 16 September 2025

  • MSC : 12E20, 12K99, 16Y20, 20N20

  • The quotient hyperfield is a landmark on the borderline of fields and hyperfields. In this paper, which is the second part of our previously published paper, all the hyperfields of order 7 are constructed, enumerated, and presented. While doing so, an important family of 7-element canonical hypergroups was revealed. The study of these hyperfields proved the existence of both quotient and non-quotient ones among them. Their construction became feasible because it is based on a new definition of the hyperfield with less axioms, which is introduced in this paper following our proof that the axiom of reversibility can derive from the remaining axioms of the hyperfield. Hence, the processing power needed for a computer to test whether a structure is a hyperfield or not, is much less. This paper also presents properties and provides examples of skew hyperfields, strongly canonical hyperfields/hyperrings, and superiorly canonical hyperfields/hyperrings that wrap up and complete the previously published first part's conclusions and results.

    Citation: Christos G. Massouros, Gerasimos G. Massouros. On the borderline of fields and hyperfields, part Ⅱ – Enumeration and classification of the hyperfields of order 7[J]. AIMS Mathematics, 2025, 10(9): 21287-21421. doi: 10.3934/math.2025951

    Related Papers:

  • The quotient hyperfield is a landmark on the borderline of fields and hyperfields. In this paper, which is the second part of our previously published paper, all the hyperfields of order 7 are constructed, enumerated, and presented. While doing so, an important family of 7-element canonical hypergroups was revealed. The study of these hyperfields proved the existence of both quotient and non-quotient ones among them. Their construction became feasible because it is based on a new definition of the hyperfield with less axioms, which is introduced in this paper following our proof that the axiom of reversibility can derive from the remaining axioms of the hyperfield. Hence, the processing power needed for a computer to test whether a structure is a hyperfield or not, is much less. This paper also presents properties and provides examples of skew hyperfields, strongly canonical hyperfields/hyperrings, and superiorly canonical hyperfields/hyperrings that wrap up and complete the previously published first part's conclusions and results.



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