Representations of Domains via CF-approximation Spaces

TL;DR

This paper introduces CF-approximation spaces to represent domains, proving their properties and establishing an equivalence between these spaces and continuous domains through categorical methods.

Contribution

It defines CF-approximation spaces and CF-closed sets, proving their correspondence with continuous domains and establishing categorical equivalences.

Findings

CF-closed sets form a continuous domain

Every continuous domain is isomorphic to CF-closed sets of some CF-approximation space

Category of CF-approximation spaces is equivalent to that of continuous domains

Abstract

Representations of domains mean in a general way representing a domain as a suitable family endowed with set-inclusion order of some mathematical structures. In this paper, representations of domains via CF-approximation spaces are considered. Concepts of CF-approximation spaces and CF-closed sets are introduced. It is proved that the family of CF-closed sets in a CF-approximation space endowed with set-inclusion order is a continuous domain and that every continuous domain is isomorphic to the family of CF-closed sets of some CF-approximation space endowed with set-inclusion order. The concept of CF-approximable relations is introduced using a categorical approach, which later facilitates the proof that the category of CF-approximation spaces and CF-approximable relations is equivalent to that of continuous domains and Scott continuous maps.