On the Borderline of Fields and Hyperfields

Christos G. Massouros () and Gerasimos G. Massouros ()
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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

Mathematics, 2023, vol. 11, issue 6, 1-35

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 ?

Keywords: fields; hyperfields; rings; hyperrings; multiplicative subgroups; hypergroups; canonical hypergroups (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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