Information systems for continuous posets

Abstract

The method of information systems is extended from algebraic posets to continuous posets by taking a set of tokens with an ordering that is transitive and interpolative but not necessarily reflexive. This develops the results of Raney on completely distributive lattices and of Hoofman on continuous Scott domains, and also generalizes Smyth's “R-structures”. Various constructions on continuous posets have neat descriptions in terms of these continuous information systems; here we describe Hoffmann-Lawson duality (which could not be done easily with R-structures) and Vietoris power locales. We also use the method to give a partial answer to a question of Johnstone's: in the context of continuous posets. Vietoris algebras are the same as localic semilattices.