Maximal $d$-spectra via Priestley duality

TL;DR

This work leverages Priestley duality to analyze maximal $d$-spectra of arithmetic frames by translating nuclei to nuclear subsets on the dual space. It establishes a precise correspondence between the spectrum $\max L_d$ and the dual object $\min Y_d$, and shows how the regularity and local-structural properties of $L_d$ reflect in the topology of these dual spaces, including soberification and localic aspects. A central achievement is resolving an open problem by constructing a unit-containing arithmetic frame with a non-Hausdorff $\max L_d$, and the paper provides a sharp criterion for when $\max L_d$ is Hausdorff in terms of stable local compactness. The results deepen the connection between algebraic-frame properties and Priestley-space topology, and raise new questions about compactness, sobriety, and the realization of spaces as duals of maximal $d$-spectra.

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.

Figures (3)

• Figure 1: An arithmetic L-space in which $Y_d \neq \max Y$.
• Figure 2: The Priestley space of an arithmetic frame whose maximal $d$-spectrum is not Hausdorff.
• Figure 3: The Priestley space of an arithmetic frame without a unit whose maximal $d$-spectrum is compact Hausdorff

Theorems & Definitions (128)

• Theorem 2.1: Priestley duality
• Remark 2.2
• Definition 2.3
• Remark 2.4
• Theorem 2.5: Pultr-Sichler duality
• Lemma 2.6
• Definition 2.7
• Theorem 2.8: see, e.g., BezhanishviliMelzer2022b
• Remark 2.9
• Definition 2.10