On the Borderline of Fields and Hyperfields
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, a...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-03-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/6/1289 |
| _version_ | 1797610386926600192 |
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| author | Christos G. Massouros Gerasimos G. Massouros |
| author_facet | Christos G. Massouros Gerasimos G. Massouros |
| author_sort | Christos G. Massouros |
| collection | DOAJ |
| description | 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 <i>F</i>’s multiplicative subgroup <i>G</i> from itself generate <i>F</i>? 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 <i>F</i>’s multiplicative subgroup <i>G</i> from itself give all the elements of the field <i>F</i>, except the ones of its multiplicative subgroup <i>G</i>? |
| first_indexed | 2024-03-11T06:13:40Z |
| format | Article |
| id | doaj.art-c126468139e94ca5a495b68b05951831 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T06:13:40Z |
| publishDate | 2023-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-c126468139e94ca5a495b68b059518312023-11-17T12:26:31ZengMDPI AGMathematics2227-73902023-03-01116128910.3390/math11061289On the Borderline of Fields and HyperfieldsChristos G. Massouros0Gerasimos G. Massouros1Core Department, Euripus Campus, National and Kapodistrian University of Athens, GR 34400 Euboia, GreeceSchool of Social Sciences, Hellenic Open University, GR 26335 Patra, GreeceThe 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 <i>F</i>’s multiplicative subgroup <i>G</i> from itself generate <i>F</i>? 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 <i>F</i>’s multiplicative subgroup <i>G</i> from itself give all the elements of the field <i>F</i>, except the ones of its multiplicative subgroup <i>G</i>?https://www.mdpi.com/2227-7390/11/6/1289fieldshyperfieldsringshyperringsmultiplicative subgroupshypergroups |
| spellingShingle | Christos G. Massouros Gerasimos G. Massouros On the Borderline of Fields and Hyperfields Mathematics fields hyperfields rings hyperrings multiplicative subgroups hypergroups |
| title | On the Borderline of Fields and Hyperfields |
| title_full | On the Borderline of Fields and Hyperfields |
| title_fullStr | On the Borderline of Fields and Hyperfields |
| title_full_unstemmed | On the Borderline of Fields and Hyperfields |
| title_short | On the Borderline of Fields and Hyperfields |
| title_sort | on the borderline of fields and hyperfields |
| topic | fields hyperfields rings hyperrings multiplicative subgroups hypergroups |
| url | https://www.mdpi.com/2227-7390/11/6/1289 |
| work_keys_str_mv | AT christosgmassouros ontheborderlineoffieldsandhyperfields AT gerasimosgmassouros ontheborderlineoffieldsandhyperfields |