Boolean metric spaces and Boolean algebraic varieties

TL;DR

This paper introduces Boolean metric spaces and algebraic varieties over certain rings, establishing a correspondence that simplifies the study of polynomial solutions in these algebraic structures.

Contribution

It defines CFG-spaces and demonstrates their equivalence to algebraic varieties over CFG-rings, advancing the understanding of polynomial maps in these rings.

Findings

Boolean metric spaces characterize polynomial maps in CFG-rings

Algebraic varieties correspond to CFG-spaces in these rings

The study of polynomial solutions reduces to Boolean metric space analysis

Abstract

The concepts of Boolean metric space and convex combination are used to characterize polynomial maps in a class of commutative Von Neumann regular rings including Boolean rings and p-rings, that we have called CFG-rings. In those rings, the study of the category of algebraic varieties (i.e. sets of solutions to a finite number of polynomial equations with polynomial maps as morphisms) is equivalent to the study of a class of Boolean metric spaces, that we call here CFG-spaces.