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On the asymptotic behavior of finite hyperfields

Tuong Le, Chayim Lowen

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.

On the asymptotic behavior of finite hyperfields

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.
Paper Structure (7 sections, 34 theorems, 67 equations, 1 figure)

This paper contains 7 sections, 34 theorems, 67 equations, 1 figure.

Table of Contents

  1. Introduction and summary of results
  2. Hyperfields, pastures and hexagons
  3. A hyperfield lottery
  4. Proofs of results
  5. Beyond the commutative case
  6. Products of hyperfields
  7. The sparsity of finite skew hyperfields

Key Result

Theorem 1.2

[Thm]thm:main For a finite abelian group $G$ and $\epsilon \in G$ of order at most $2$, let $\mathcal{H}(G, \epsilon)$ be the set of hyperfields with underlying group $G$ in which $-1$ is given by $\epsilon$---up to isomorphism preserving $\epsilon$. Let also $\mathcal{I}(G, \epsilon)$ be the subset

Figures (1)

  • Figure 1: A hexagon of fundamental pairs

Theorems & Definitions (94)

  • Conjecture 1.1: Jin
  • Theorem 1.2
  • Corollary 1.3
  • Definition 1.4
  • Remark 1.5
  • Definition 1.6
  • Remark 1.7
  • Theorem 1.8
  • Corollary 1.9
  • Corollary 1.10
  • ...and 84 more