Representations of Domains via CF-approximation Spaces
This work provides a new representation framework for continuous domains in domain theory, which is incremental as it builds on existing mathematical structures.
The paper tackles the problem of representing continuous domains by introducing CF-approximation spaces and CF-closed sets, proving that every continuous domain is isomorphic to such a family under set-inclusion order, and establishing a categorical equivalence between CF-approximation spaces and continuous domains.
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.