$\mathcal{T}$-semiring 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. 4.1. Fractions 10 4.2. Integral extensions 10 4.3. Hilbert Nullstellensatz 12 5. Growth in semialgebras 13 5.1. Growth in a pair 13 References 14