On the borderline of fields and hyperfields, part II -- Enumeration and classification of the hyperfields of order 7

TL;DR

This paper classifies all hyperfields of order 7, introduces a simplified hyperfield definition reducing axioms, and reveals new hyperfield families, enhancing understanding of hyperfield structures and their computational testing.

Contribution

It constructs and enumerates hyperfields of order 7, introduces a minimal axiom hyperfield definition, and identifies new hyperfield families, advancing hyperfield theory.

Findings

Existence of quotient and non-quotient hyperfields of order 7

Introduction of a simplified hyperfield definition with fewer axioms

Discovery of new hyperfield families and properties

Abstract

The quotient hyperfield is a landmark on the borderline of fields and hyperfields. In this paper, which is the second part of our previously published paper, all the hyperfields of order 7 are constructed, enumerated and presented, in the course of which an important family of 7-element canonical hypergroups is revealed. The study of these hyperfields proved the existence of both quotient and non-quotient ones among them. Their construction became feasible because it is based on a new definition of the hyperfield with fewer axioms, which is introduced in this paper following our proof that the axiom of reversibility can derive from the other axioms of the hyperfield. Hence, the processing power needed for a computer to test whether a structure is a hyperfield or not is much less. This paper also proves properties and contains examples of skew hyperfields, strongly canonical hyperfields/hyperrings and superiorly canonical hyperfields/hyperrings that wrap up and complete the previously published conclusions and results of its first part.