Partial information systems and the Smyth powerdomain (Q2905563)

The primary goal of this paper is to provide a representation of the Smyth powerdomain in terms of order-convex substructures in its underlying information system. To accomplish this goal, the author introduces \textit{partial information systems} as a variation on the well-known concept of information system. In this variation, the author recasts the notion of entailment as a preorder and introduces the concept of a \textit{frontier} within the consistency predicate. Partial information systems are intended to represent ``incomplete'' knowledge of the consistency predicate within an information system, and the frontier represents the ``limit'' of knowledge. The author uses this concept to provide a semantic understanding of upper sets in the consistency predicate of a ``complete'' information system -- these sets are identified with \textit{saturated} (order-convex) partial sub-information systems whose frontiers in turn serve as generators for the upper set. The author concludes the paper by proving that the join semilattice of saturated partial sub-information systems having finite frontiers in an information system is order-isomorphic to the poset of nonempty, compact Scott-open subsets of the corresponding domain. The desired representation is then a consequence of a known result providing a dual order-isomorphism between this poset and the Smyth powerdomain.