Domains via a rough set theoretic approach

This paper proposes a rough set theoretic approach to Domain theory, thereby establishing a direct connection between Rough Set Theory and Domain Theory. With a rough set theoretic mind-set, we tailor-made new approximation operators specially suited for Domain Theory. Our proposed approach not only offers a fresh perspective to existing concepts and results in Domain Theory through the rough set theoretic lens, but also reveals ways to establishing novel domain-theoretic results. For instance, (1) the well-known interpolation property of the way-below relation on a continuous poset is equivalent to the idempotence of a certain set-operator; (2) the continuity of a poset can be characterized by the coincidence of the Scott closure operator and the upper approximation operator induced by the way below relation; and as a result, (3) a new characterization of Scott closure is obtained. Additionally, we show how, to each approximating relation, an associated order-compatible topology can be defined in such a way that for the case of a continuous poset the topology associated to the way-below relation is exactly the Scott topology. A preliminary investigation is carried out on this new topology.