Kummer structures

TL;DR

This paper introduces the concept of Kummer structures derived from abelian groups quotiented by negation, characterizes their properties, and classifies all such structures through specific axioms and constructions.

Contribution

It defines Kummer structures in terms of a map on unordered pairs and classifies them using axioms, linking them to known group quotients and involutions.

Findings

Every Kummer structure corresponds to a unique abelian group quotient.

Some structures arise from 2-torsion groups with involutions.

The paper provides axioms characterizing all Kummer structures.

Abstract

Suppose we take an abelian group G and quotient it by the action of negation. What structure does the quotient K inherit from the group structure of G? We describe this structure (which we call the Kummer of G) in terms of a map from the set of unordered pairs of elements of K to itself. We propose some axioms that hold for such structures, and show that every structure satisfying those axioms either is the Kummer of a unique group, or comes from one other construction, the quotient of a 2-torsion group by an involution.