Characterize when z-Spec(C(T)) is Esakia (equivalently, when Coz(T) is a Heyting algebra)

Characterize completely regular spaces T for which the spectral subspace z-Spec(C(T)) of prime z-ideals of C(T) is an Esakia space; equivalently, determine necessary and sufficient conditions on T under which the lattice of cozero sets Coz(T) forms a Heyting algebra.

Background

The authors establish many results on pseudocomplementation and Stone/Hnovan properties for K(Spec C(T)) and related structures, as well as equivalences for inverse spectra. For z-Spec(C(T)), they note that Esakia-ness occurs frequently—for instance, when all open sets are cozero sets (e.g., in perfectly normal spaces)—but a full characterization is not available.

Since K(z-Spec C(T)) is (order-dual to) Coz(T), z-Spec(C(T)) being Esakia is equivalent to Coz(T) being a Heyting algebra. Despite several sufficient conditions, a complete necessary-and-sufficient description remains unknown.

References

We also do not have a general characterization of when z-Spec(C(T)) is Esakia (equivalently: Coz(T) is a Heyting algebra).

— Pseudocomplementation in rings of continuous functions  (2603.28165 - Bezhanishvili et al., 30 Mar 2026) in Section 5.11, paragraph following the Open problem (immediately before References)