Maximal $d$-spectra via Priestley duality

TL;DR

This paper employs Priestley duality to analyze maximal $d$-spectra of arithmetic frames, providing conditions for their topological properties and resolving an open problem about non-Hausdorff spectra.

Contribution

It introduces a novel application of Priestley duality to study maximal $d$-spectra, including a construction of a frame with a non-Hausdorff spectrum.

Findings

Characterization of when the maximal $d$-spectrum is compact or Hausdorff

Necessary and sufficient conditions for spectral properties

Construction of a frame with a non-Hausdorff maximal $d$-spectrum

Abstract

We use Priestley duality as a new tool to study maximal $d$-spectra of arithmetic frames, both with and without units. We pay special attention to when the maximal $d$-spectrum is compact or Hausdorff. Various necessary and sufficient conditions are given, including a construction of an arithmetic frame with a unit whose maximal $d$-spectrum is not Hausdorff, thus resolving an open problem in the literature.