Domains, Information Frames, Rough Sets: An Equivalence of Categories

TL;DR

This paper generalizes Scott's information systems by relativizing the consistency predicate, establishing an equivalence between the category of information frames and domains, and refining the relationship with CF-approximation spaces.

Contribution

It introduces a generalized framework for information systems that captures all continuous domains and proves categorical equivalences with related structures.

Findings

Category of information frames is equivalent to the category of domains

Relativized consistency predicates refine Scott's original definition

Equivalence between CF-approximation spaces and domains is refined

Abstract

A generalization of Scott's information systems~\cite{sco82} is presented that captures exactly all continuous domains. The global consistency predicate in Scott's definition is relativized. Now, for every atomic statement, there is a consistency predicate that states which finite sets of statements express information that is consistent with the given statement. The category of information frames is shown to be equivalent to the category of domains. Moreover, the relationship with CF-approximation spaces introduced by Wu and Xu~\cite{wx23} is studied. The corresponding category is also shown to be equivalent with the category of information frames. This research achieves a refinement of the equivalence result of Wu and Xu of the category of CF-approximation spaces with the category of domains.