On the choice of topological dual of a lattice

<div> We consider a unified setting for topological dual structures of lattices based on Priestley duality for adjunctions between distributive lattices and the associated discrete duality. We show that each of the three existing choices of duals are natural and how they are related using the Priestley structure. In this setting, the Hofmann-Mislove-Stralka dual space is dual to an adjunction whose lattice of fixed points is isomorphic to the original lattice and it is the largest such in a natural sense. We show that the two other fundamentally different choices of dual structure of a lattice are the least subpolarities of this polarity having the same fixed points as the full polarity, respectively, among Priestley compatible polarities and among upward order compatible polarities. In this way we obtain, respectively, the Gehrke-van Gool dual space and the Hartung variant of Urquhart's dual. While it is well known how the lattices dual to these topological polarities are related, we obtain here a purely topo-relational description of the relationship between these structures. Finally, our results allow us to give a simpler description of the objects in the Gehrke-van Gool duality. </div>