# Dualities of domains

## Description

The basic mathematical structures used in denotational semantics to model programming languages are algebraic posets. A more general notion is that of a continuous poset. It is known that when equipped with the Scott topology, a continuous poset is locally compact and has a basis of open filters. It turns out that these two conditions are sufficient to ensure that a poset is continuous. In general, if a poset has only a basis of open filters, we obtain that the Scott open interior map from its lattice of upper sets to its lattice of open sets preserves finite union, i.e., $(A\ \cup\ B)\sp\circ = A\sp\circ\ \cup\ B\sp\circ$ if A and B are upper sets Spectral theory studies the relation between a topological space and its lattice of closed (or open) sets. It is known that when we apply this theory to those spaces which arise from the Scott topology of a continuous dcpo (directed complete poset), a duality between continuous dcpo's and completely distributive lattices is established and is known as the Lawson Duality. If we impose the condition that each continuous dcpo has a bottom, and consider only the lattice of non-empty closed sets, we obtain an equivalence between the category ${\bf CONT}\sb\perp$ of continuous dcpo's with bottom and Scott continuous maps as morphisms and the category ${\bf CDL}\sbsp{1}{\prime}$ of completely distributive lattices with maps preserving all non-empty suprema and sup-primes as morphisms. If the requirement of preserving sup-primes on the morphisms ${\bf CDL}\sbsp{1}{\prime}$ is dropped, we obtain a larger category CDL, and the equivalence above gives an adjunction which generalizes the Hoare power domain The Smyth power domain is another important power domain construction. Given an algebraic dcpo, its Smyth power domain is characterized as the algebraic semilattice of non-empty Scott compact and saturated subsets of the underlying dcpo. We generalize this to non-algebraic dcpo's and characterize their semilattices of non-empty Scott compact and saturated subsets. First, an equivalence is established between the category QCONT of quasi-continuous dcpo's with Scott continuous maps as morphisms and the category CSL' of the continuous semilattices characterized by the property that the way-below relation is multiplicative and prime separating, with Scott continuous semilattice prime-preserving maps as morphisms. Then several of its subdualities are studied. Finally the requirement that the morphisms preserve primes is dropped, then an adjunction is obtained which generalizes the Smyth power domain By combining the two equivalences thus obtained, we obtain an equivalence between ${\bf CDL}\sbsp{1}{\prime}$ and ${\bf CSL}\sbsp{\rm c}{\prime},$ which is a subcategory of CSL$\sp\prime$ equivalent to ${\bf CONT}\sb\perp.$ This reveals a very interesting relationship between two of the very important power domains