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Spatiality of the ideal d-frame for arbitrary d-lattices

Determine whether the d-frame of ideals Idl(𝓛) is spatial for every d-lattice 𝓛; that is, establish whether Idl(𝓛) is isomorphic to the d-frame of open sets dO(X, Ο„+, Ο„βˆ’) for some bitopological space (X, Ο„+, Ο„βˆ’).

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Background

For a d-lattice 𝓛, the construction Idl(𝓛) yields a d-frame. A d-frame is spatial if it arises as the d-frame of open sets of a bitopological space. The authors prove spatiality of Idl(𝓛) when 𝓛 is a d-Boolean algebra, but the general case remains unresolved.

Establishing spatiality in full generality would connect the ideal construction to bitopological spaces for arbitrary d-lattices, paralleling classical frame–space dualities.

References

We do not know whether the d-frame $\idl\mathcal{L}$ is spatial for every d-lattice $\mathcal{L}$.

— d-Boolean algebras and their bitopological representation (2505.17806 - Yang et al., 23 May 2025) in Section 5 (Spectra of d-Boolean algebras), following Proposition \ref{spectrum as composite}