Then, we provide a general framework for such dualities, using so-called R-invariant subset selections Z for (partially) ordered sets. These selections extend the concept of subset systems proposed by Wright, Wagner and Thatcher in the late seventies and prove quite useful in various fields of order theory, topology, algebra and computer sciences. The guiding idea is that practically all known dualities of the order-topological type involve various notions of primeness on the one hand and certain generalized ideal systems on the other hand. The resulting duality between arbitrary ordered sets and Z-generated complete lattices, having a join-dense subset of Z-(join-)prime elements, is established by the Z-ideal (or closed set) functor in the one direction and the functorial restriction to the Z-join spectra, consisting of all Z-prime elements, in the other direction.

Even more effective (and closely related to the ``Fundamental Duality'' due to Banaschewski and Bruns) is what we call a symmetric duality, involving two subset selections, X and Z, instead of one. Here, on the ``topological side'', one considers the category ZS X of all T₀ closure spaces having a basis of X-prime open sets and exactly the point closures as Z-prime closed sets (Z-soberness). Morphisms are maps such that inverse images of X-prime open sets are X-prime open. On the ``order-theoretical side'', one takes the category ZC X of Z-generated lattices whose dual is X-generated. Morphisms preserve arbitrary suprema, X-infima and Z-spectra. A central observation is now that order-theoretical adjunction between the involved morphisms and ``comorphisms'' entails a dual adjunction between the respective categories of lattices. Using that fact, one obtains a dual isomorphism between the categories ZCX and XC Z, just by passing to the adjoints and then turning domain and codomain upside down. Composing this dual isomorphism with the aforementioned equivalences established by the spectrum and closed set functors, respectively, one arrives at a duality between the categories ZSX and XS Z. It turns out that many known, but also some interesting new dualities are special instances of that construction. For example, the classical Stone duality, extended to the one between distributive join-semilattices and sober spaces with compact-open bases, arises in that fashion if X is the selection of finite subsets and Z is that of directed subsets. Other choices of X and Z lead, for example, to the following equivalences and dualities:

• sober spaces  -  primely generated frames  -  spatial locales

  (Dowker-Strauss, Isbell, Johnstone)

• Alexandroff spaces  -  superalgebraic lattices  -  ordered sets

  (Alexandroff, Raney)

• sober core spaces - completely distributive lattices - continuous posets

  (Erné, Hoffmann, Lawson)

In the last part, we touch upon connections between the generalized Stone Dualities and Formal Concept Analysis, a modern applied theory developed by Wille and his school. The basic notions are here contexts (consisting of two sets and a relation between them) and their concept lattices, constituted by the associated polarity (Galois connection) in the sense of Birkhoff and Ore.