Domains, Information Frames, Rough Sets: An Equivalence of Categories
Dieter Spreen
TL;DR
This work develops a unifying framework that links domain theory, information frames, rough sets, and CF-approximation spaces. By introducing information frames and approximable mappings, it proves an equivalence between the category of information frames and the category of domains with Scott-continuous functions, extending Scott's information systems via a relativized consistency predicate. It further connects these frames to Wu and Xu's CF-approximation spaces and shows that CF-approximation spaces and CF-approximable relations are categorically equivalent to information frames, yielding a refined chain of equivalences. The results provide modular constructions and natural isomorphisms that translate between domains, information frames, and CF-approximation spaces, clarifying the relationships among these formalisms and enabling algebraic/pointed/topological refinements. Overall, the paper presents a canonical, category-theoretic integration of domain theory and rough-set-inspired representations, with strong implications for semantic modeling and related logics.
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.