Formal Concept Analysis (FCA) has been proven to be an effective method of restructuring complete lattices and various algebraic domains. In this paper, the notion of contractive mappings over formal contexts is proposed, which can be viewed as a generalization of interior operators on sets into the framework of FCA. Then, by considering subset-selections consistent with contractive mappings, the notions of attribute continuous formal contexts and continuous concepts are introduced. It is shown that the set of continuous concepts of an attribute continuous formal context forms a continuous domain, and every continuous domain can be restructured in this way. Moreover, the notion of F-morphisms is identified to produce a category equivalent to that of continuous domains with Scott continuous functions. The paper also investigates the representations of various subclasses of continuous domains including algebraic domains and stably continuous semilattices.

In this paper, we introduce the notion of consistent F-augmented contexts by adding a special family of finite subsets into the structure of a formal context, which essentially establishes the basis of the representation of general algebraic domains. In particular, we investigate the association rule systems which are derived from the consistent F-augmented contexts and propose the notion of formal association rule systems. By the notion of antecedent connections, we obtain the equivalence between the category of formal association rule systems and that of algebraic domains, which demonstrates that the proposed notion of formal association rule systems provides a concrete approach to representing algebraic domains.

The introduction of information system homomorphisms has made a substantial contribution to attribute reduction. However, the efforts made on homomorphisms are far from sufficient. This paper further investigates homomorphisms between covering approximation spaces. First, we introduce the concepts of upper and lower homomorphisms as well as homomorphisms in order to study the relationship between covering approximation spaces. Then we present the notions of covering approximation subspaces and product spaces. We also compress covering approximation spaces and covering information systems with the aim of attribute reduction. Afterwards, by utilizing the compressions of the original spaces and systems we compress the dynamic covering approximation spaces and dynamic covering information systems. Several illustrative examples are employed to demonstrate that the homomorphisms provide an effective approach for compressing covering approximation spaces and covering information systems.