Classification of Krasner Hyperfields of Order 4

Iranmanesh, Mahdeyeh; Jafarpour, M.; Aghabozorgi, H.; Zhan, Jianming

doi:10.1007/s10114-020-8282-z

Cited by 2 publications

(5 citation statements)

References 13 publications

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“…A large number of papers has been published on the hyperfields and hyperrings, starting from the pioneer work of J. Mittas [37][38][39][40][41][42][43][44] and continuing with a plenitude of researchers, such as Ch. Massouros [29,35,[45][46][47][48][49][50][51], A. Nakassis [36], G. Massouros [50][51][52][53][54], R. Rota [55,56], S. Jančic-Rašović [57][58][59], I. Cristea [58][59][60][61][62][63][64], H. Bordbar [59][60][61], M. Kankaraš [62], V. Vahedi et al [63][64][65], M. Jafarpour et al [63][64][65][66], A. Connes and C. Consani [67,68], O. Viro [69,70], R. Ameri, M. Eyvazi and S. Hoskova-Mayero...…”

Section: The Quotient Hyperfield/hyperringmentioning

confidence: 99%

Section: Classification Of Finite Hyperfields Into Quotient and Non-q...mentioning

confidence: 99%

“…The enumeration of certain finite hyperfields has been conducted in several papers [66,71,73,77]. Paper [66] deals with hyperfields of order less than or equal to 4, [73,77] deals with hyperfields of order less than or equal to 5, and [71] deals with hyperfields of order less than or equal to 6. In [71], R. Ameri, M. Eyvazi, and S. Hoskova-Mayerova make a thorough check of the isomorphism of these hyperfields to the quotient hyperfields using conclusions from the papers [46][47][48][95][96][97].…”

Section: Classification Of Finite Hyperfields Into Quotient and Non-q...mentioning

confidence: 99%

“…Hyperfields of order 3 have two non-zero elements. There are five isomorphism classes of these hyperfields [66,71,73,77]. The trivial hyperfield Z 3 is the first of them.…”

Section: Hyperfields Of Ordermentioning

confidence: 99%

“…There are 7 isomorphism classes of hyperfields of order 4, as they have been enumerated in [66,71,73,77]. These consist of the Galois field GF [2 2 ], 4 classes of quotient hyperfields, and 2 classes of non-quotient hyperfields.…”

Section: Hyperfields Of Ordermentioning

confidence: 99%

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Hypergroup Theory and Algebrization of Incidence Structures

2023

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Section: The Quotient Hyperfield/hyperringmentioning

confidence: 99%

Section: Classification Of Finite Hyperfields Into Quotient and Non-q...mentioning

confidence: 99%

Section: Classification Of Finite Hyperfields Into Quotient and Non-q...mentioning

confidence: 99%

“…Hyperfields of order 3 have two non-zero elements. There are five isomorphism classes of these hyperfields [66,71,73,77]. The trivial hyperfield Z 3 is the first of them.…”

Section: Hyperfields Of Ordermentioning

confidence: 99%

Section: Hyperfields Of Ordermentioning

confidence: 99%

See 3 more Smart Citations

Hypergroup Theory and Algebrization of Incidence Structures

2023

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On the Borderline of Fields and Hyperfields

Massouros

2023

Mathematics

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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?

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Classification of Krasner Hyperfields of Order 4

Cited by 2 publications

References 13 publications

Hypergroup Theory and Algebrization of Incidence Structures

Hypergroup Theory and Algebrization of Incidence Structures

On the Borderline of Fields and Hyperfields

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