The Priestley spaces of Examples 8.1 and 8.11 in Maximal d-spectra via Priestley duality with Guram Bezhanishvili and Papiya Bhattacharjee (PDF, DOI).
Example 8.1 (Fig. 2). The Priestley space of an arithmetic frame with a unit whose maximal d-spectrum is not Hausdorff, answering an open question in the negative. It is βℕ ⊔ Yω, where ℕ = X0 ∪ X1 ∪ … Each yi lies below cl Xi, and every yi ≤ ω. Closedness of the order forces ω ≤ Xω*.Example 8.11 (Fig. 3). The Priestley space of an arithmetic frame without a unit whose maximal d-spectrum is compact Hausdorff (min Yd is empty). It shows that Theorem 7.7 needs the assumption that Nd is d-initial. The space is that of Example 8.1 with the order redefined: y0 ≥ y1 ≥ … → ω, each yi below cl Xi. Closedness forces ω ≤ yi and ω ≤ Xω*.