$\mathcal{T}$-semiring pairs

TL;DR

This paper introduces a broad axiomatic framework for algebraic pairs that unifies various structures and explores classical algebraic theorems within this new setting, including fractions, extensions, and growth.

Contribution

It develops a general theory of algebraic pairs that generalizes multiple structures and adapts classical theorems to this framework, reducing reliance on negation.

Findings

Unified classical theorems in the new framework

Extended notions of fractions and integral extensions

Studied growth properties in algebraic pairs

Abstract

We develop a general axiomatic theory of algebraic pairs, which simultaneously generalizes several algebraic structures, in order to bypass negation as much as feasible. We investigate several classical theorems and notions in this setting including fractions, integral extensions, and Hilbert's Nullstellensatz. Finally, we study a notion of growth in this context.