On the asymptotic behavior of finite hyperfields

TL;DR

This paper proves that nearly all finite hyperfields are not quotients of fields, extending previous results, and analyzes their additive properties and enumeration on finite abelian groups.

Contribution

It extends the non-quotient property of finite hyperfields to all orders using probabilistic methods and provides asymptotic counts for hyperfields on finite abelian groups.

Findings

Almost all finite hyperfields are non-quotients of fields.

In almost every finite hyperfield, the sum of four or more nonzero elements contains zero.

Provides asymptotic enumeration of hyperfields on finite abelian groups.

Abstract

Hobby has recently shown that almost all finite hyperfields of even order fail to be the quotient of a field. Using a probabilistic argument, we extend this result to all orders: a finite hyperfield is almost always non-quotient. This confirms a conjecture of Baker--Jin. We show that in almost every finite hyperfield the sum of any four or more nonzero elements contains 0. We also give a precise asymptotic for the number of finite hyperfields on a given finite abelian group.