### $\mathcal{Z}$-distributive function lattices

It is known that for a nonempty topological space $X$ and a nonsingleton complete lattice $Y$ endowed with the Scott topology, the partially ordered set $[X,Y]$ of all continuous functions from $X$ into $Y$ is a continuous lattice if and only if both $Y$ and the open set lattice $\mathcal{O}X$ are continuous lattices. This result extends to certain classes of $\mathcal{Z}$-distributive lattices, where $\mathcal{Z}$ is a subset system replacing the system $\mathcal{D}$ of all directed subsets (for which the $\mathcal{D}$-distributive complete lattices are just the continuous...