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.
Matroids over hyperfields
1 Pith paper cite this work. Polarity classification is still indexing.
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abstract
We present an algebraic framework which simultaneously generalizes the notion of linear subspaces, matroids, valuated matroids, and oriented matroids. We call the resulting objects matroids over hyperfields. In fact, there are (at least) two natural notions of matroid in this context, which we call weak and strong matroids. We give "cryptomorphic" axiom systems for such matroids in terms of circuits, Grassmann-Plucker functions, and dual pairs, and establish some basic duality theorems. We also show that if F is a doubly distributive hyperfield then the notions of weak and strong matroid over F coincide.
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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.