Representations of FS-domains and BF-domains via FS-approximation Spaces

TL;DR

This paper introduces FS-approximation spaces and demonstrates their ability to represent FS-domains and BF-domains, establishing isomorphisms and categorical equivalences without relying on CF-approximable relations.

Contribution

It provides new topological representations of FS- and BF-domains via FS-approximation spaces and introduces a method to represent BF-domains without CF-approximable relations.

Findings

FS-approximation spaces represent FS-domains and BF-domains as CF-closed sets.

Every FS- or BF-domain is order isomorphic to a collection of CF-closed sets.

The category of FS- and BF-domains is equivalent to that of FS-approximation spaces.

Abstract

In this paper, concepts of (topological) FS-approximation spaces are introduced. Representations of FS-domains and BF-domains via (topological) FS-approximation spaces are considered. It is proved that the collection of CF-closed sets in an FS-approximation space (resp., a topological FS-approximation space) endowed with the set-inclusion order is an FS-domain (resp., a BF-domain) and that every FS-domain (resp., BF-domain) is order isomorphic to the collection of CF-closed sets of some FS-approximation space (resp., topological FS-approximation space) endowed with the set-inclusion order. The concept of topological BF-approximation spaces is introduced and a skillful method without using CF-approximable relations to represent BF-domains is given. It is also proved that the category of FS-domains (resp., BF-domains) with Scott continuous maps as morphisms is equivalent to that of FS-approximation spaces (resp., topological FS-approximation spaces) with CF-approximable relations as morphisms.