Blocks in Finite Hyperfields

Abstract

This paper studies the structure of finite hyperfields $H$, and finds a subtle pattern in their addition operation. Consider the class $\mathcal{H}$ of all hyperfields with a given multiplicative group on $H^\times = H - \{0\}$ and given value of $-1$. Then the addition of hyperfields in this class is determined by the set of pairs $(x,y)$ with $y \in x+1$ for $x,y \in H^\times$. There are blocks of such pairs, where $(x_0,y_0)$ and $(x_1,y_1)$ are in the same block iff every hyperfield with $y_0 \in x_0 + 1$ also has $y_1 \in x_1 + 1$. The theory of these blocks is developed, they can easily be computed without using hyperfields. Exploiting this theory of blocks would greatly speed up future computer searches for small hyperfields. The theory of blocks is then used to show that the number of nonquotient hyperfields of size $n$ grows exponentially with $n$, and that for even $n$ most hyperfields are nonquotient. A final application shows that a large class of finite hyperfields has the FETVINS property, meaning that systems of linear homogeneous equations with fewer equations than variables always have nontrivial solutions.