Domains via approximation operators

TL;DR

This paper introduces new approximation operators inspired by rough set theory tailored for domain theory, providing novel characterizations of properties like continuity and the Scott topology.

Contribution

It develops approximation operators that connect rough set concepts with domain theory, leading to new characterizations and a new topology related to the way-below relation.

Findings

Interpolation property linked to idempotence of a set-operator

Continuity characterized by Scott closure and upper approximation coincidence

Meet-continuity derived from topological closure properties

Abstract

In this paper, we tailor-make new approximation operators inspired by rough set theory and specially suited for domain theory. Our approximation operators offer a fresh perspective to existing concepts and results in domain theory, but also reveal 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; (3) meet-continuity can be established from a certain property of the topological closure operator. 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.