Dual to bounded distributive lattices are Priestley spaces, and that for Heyting algebras are Esakia spaces. Do we have such results for the class of lattices which are not bounded in general, preferably for distributive lattices or contrapositionally complemented algebras (the class of algebra corresponding to minimal logic)?
My understanding: The set of clopen sets of a Priestley space forms a bounded distributive lattice. Now, empty set and $X$ are always the smallest and largest clopen sets respectively in any topological space over $X$. Thus the lattice of the set of clopen sets will always be bounded. So, just taking the set of clopen sets will not work for unbounded case.