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

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.