Noetherian theories

TL;DR

This paper introduces the concept of Noetherian theories, a strengthening of equationality, and demonstrates that the theory of proper pairs of algebraically closed fields is Noetherian in any characteristic.

Contribution

It formalizes the notion of Noetherian theories and proves the Noetherianity of proper pairs of algebraically closed fields across all characteristics.

Findings

Noetherianity is a stronger property than equationality.

Proper pairs of algebraically closed fields are Noetherian in any characteristic.

The topology defined by formula instances is Noetherian.

Abstract

A first-order theory is Noetherian with respect to the collection of formulae $\mathcal{F}$ if every definable set is a Boolean combination of instances of formulae in $\mathcal{F}$ and the topology whose subbasis of closed sets is the collection of instances of arbitrary formulae in $\mathcal{F}$ is Noetherian. Noetherianity is a strengthening of equationality, which itself implies stability. We show the Noetherianity of the theory of proper pairs of algebraically closed fields in any characteristic.