Domains via approximation operators

Abstract

In this paper, we tailor-make new approximation operators inspired by roughset theory and specially suited for domain theory. Our approximation operatorsoffer a fresh perspective to existing concepts and results in domain theory,but also reveal ways to establishing novel domain-theoretic results. Forinstance, (1) the well-known interpolation property of the way-below relationon a continuous poset is equivalent to the idempotence of a certainset-operator; (2) the continuity of a poset can be characterized by thecoincidence of the Scott closure operator and the upper approximation operatorinduced by the way below relation; (3) meet-continuity can be established froma certain property of the topological closure operator. Additionally, we showhow, to each approximating relation, an associated order-compatible topologycan be defined in such a way that for the case of a continuous poset thetopology associated to the way-below relation is exactly the Scott topology. Apreliminary investigation is carried out on this new topology.Comment: 17 pages; 1figure, Domains XII Worksho