Finite Hyperfields of Order $n\leq 5$

TL;DR

This paper classifies all finite commutative hyperfields of orders up to 5, providing a foundational understanding of their structure and enumeration, which can aid further research in algebraic hyperstructure theory.

Contribution

It systematically enumerates all finite commutative hyperfields of orders 2 to 5, establishing their isomorphism classes and basic properties.

Findings

2 hyperfields of order 2

5 hyperfields of order 3

7 hyperfields of order 4

Abstract

In this paper, the author introduces the concept and basic properties of finite (commutative) hyperfields. Also, the author shows that, up to isomorphism, there are exactly 2 hyperfields of order 2; 5 hyperfields of order 3; 7 hyperfields of order 4; 27 hyperfields of order 5. Those hyperfields could be first hand materials for those who are doing relevant researches.