There are exactly 277 seven-element hyperfields, all built on the cyclic multiplicative group of order six, and the paper classifies which of them arise as quotients of fields.
Nontrivial solutions for homogeneous linear equations over some non-quotient hyperfields
1 Pith paper cite this work. Polarity classification is still indexing.
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abstract
We introduce a class of hyperfields which includes several constructions of non-quotient hyperfields. We then use it to partially answer a question posed by M. Baker and T. Zhang: Does a system of homogeneous linear equations with more unknowns than equations always have a nonzero solution? We also consider a class of hyperfields that was claimed in the literature to be non-quotient, and show that this is false.
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On the borderline of fields and hyperfields, part II -- Enumeration and classification of the hyperfields of order 7
There are exactly 277 seven-element hyperfields, all built on the cyclic multiplicative group of order six, and the paper classifies which of them arise as quotients of fields.