von Neumann regular Hyperrings and applications to Real Reduced Multirings

TL;DR

This paper introduces von Neumann regular hyperrings, characterizes when their structural presheaves are sheaves, and applies these concepts to the algebraic theory of quadratic forms, especially within real reduced multirings.

Contribution

It provides a first-order characterization of geometric von Neumann regular hyperrings and constructs a von Neumann regular hull for multirings, with applications to real algebraic structures.

Findings

Characterization of vNH with a structural sheaf

Construction of a von Neumann regular hull for multirings

Application to algebraic theory of quadratic forms

Abstract

A multiring ([Mar3]) is a kind of ring where is allowed the sum of two elements to be anon-empty subset of the structure instead of just one element -and an hyperring is a multiring with a strong distributive property. Thus a reduced hyperring where the prime spec is a Boolean topological space is called von Neumann regular hyperring (vNH). It is possible to associate to every such object a structural presheaf in the same way it is made with rings but there are some vNH such that this presheaf is not a sheaf. In this sense, we give a first-order characterization of vNH with a structural sheaf (geometric vNH or just GvNH) and how to transform a vNH in aGvNH -in fact, this transformation shows that the category GvNH is a reflexive subcategory of vNH. We also build a von Neumann regular hull for multirings and use this to give applications for algebraic theory of quadratic forms. More precisely, we work with Real Reduced Multiring (RRM, [Mar3]) -also known as Real Semigroup (RS, [DP1])-, a special kind of multirings that is useful to explore the real structure of rings, and show that a von Neumann hull of a RRM is again a RRM. This gives a generalization of sheafs arguments present in [DM4].