Fixed Points and Noetherian Topologies

TL;DR

This paper introduces a canonical construction for Noetherian fixed point topologies, providing a unified proof method and applying it to spaces like finite words and trees, generalizing known divisibility preorders.

Contribution

It develops a fixed point theorem for Noetherian topologies and applies it to define and analyze divisibility topologies in inductively defined spaces.

Findings

Established a minimal bad sequence argument for fixed points

Reconstructed known Noetherian topologies uniformly

Generalized divisibility preorder to well-quasi-orders

Abstract

This paper provides a canonical construction of a Noetherian least fixed point topology. While such least fixed point are not Noetherian in general, we prove that under a mild assumption, one can use a topological minimal bad sequence argument to prove that they are. We then apply this fixed point theorem to rebuild known Noetherian topologies with a uniform proof. In the case of spaces that are defined inductively (such as finite words and finite trees), we provide a uniform definition of a divisibility topology using our fixed point theorem. We then prove that the divisibility topology is a generalisation of the divisibility preorder introduced by Hasegawa in the case of well-quasi-orders.