When do CF-approximation spaces capture sL-domains

TL;DR

This paper explores the relationship between CF-approximation spaces and sL-domains, establishing conditions under which these approximation spaces precisely capture the structure of sL-domains and related classes.

Contribution

It introduces new types of approximation spaces and proves their equivalence to sL-domains, L-domains, and bc-domains, providing a categorical correspondence.

Findings

CF-closed sets form sL-domains, L-domains, or bc-domains depending on the approximation space type.

Every sL-domain, L-domain, or bc-domain is order-isomorphic to CF-closed sets of a corresponding approximation space.

Categories of sL-domains and L-domains are equivalent to categories of approximation spaces with CF-approximable relations.

Abstract

In this paper, by means of upper approximation operators in rough set theory, we study representations for sL-domains and its special subclasses. We introduce the concepts of sL-approximation spaces, L-approximation spaces and bc-approximation spaces, which are special types of CF-approximation spaces. We prove that the collection of CF-closed sets in an sL-approximation space (resp., an L-approximation space, a bc-approximation space) ordered by set-theoretic inclusion is an sL-domain (resp., an L-domain, a bc-domain); conversely, every sL-domain (resp., L-domain, bc-domain) is order-isomorphic to the collection of CF-closed sets of an sL-approximation space (resp., an L-approximation space, a bc-approximation space). Consequently, we establish an equivalence between the category of sL-domains (resp., L-domains) with Scott continuous mappings and that of sL-approximation spaces (resp., L-approximation spaces) with CF-approximable relations.