Absolute Algebra III - The saturated spectrum

TL;DR

This paper explores the structure of B1--algebras, comparing different notions of prime ideals and relating them to Deitmar's F1--schemes, advancing the understanding of algebraic geometry over the smallest characteristic 1 semifield.

Contribution

It introduces and compares two notions of prime ideals in B1--algebras and connects these concepts to the framework of F1--schemes, providing new insights into algebraic structures over characteristic 1.

Findings

Comparison of two prime ideal notions in B1--algebras

Relations established between B1--algebras and F1--schemes

Enhanced understanding of algebraic geometry in characteristic 1

Abstract

Let B1 denote the set {0,1} with the usual operations except that $1+1=1$, in other words, the smallest characteristic 1 semifield . We compare two possible analogues of the notion of prime ideal for B1--algebras. We then consider the relations between these notions and Deitmar's theory of F1--schemes.