The concept lattice restoration methods are investigated through concept reducts. First, the relationship between subsets of concept reducts and meet-irreducible concepts is established, thereby revealing the connection between concept reducts and meet-irreducible concept sets. Based on this foundation, a concept lattice restoration method is proposed in which supremums are first computed, followed by infimums, for formal concepts in concept reducts. Dually, an alternative restoration approach is presented in which infimums are first computed, then supremums. It is demonstrated that concept lattices can be restored by concept reducts using partial formal concepts, thereby achieving effective compression and storage of lattice information. This provides practical significance and potential application value for concept reduction.

Concept reduction preserving binary relation is an important research direction in formal concept analysis, which allows for knowledge discovery by selecting some formal concepts (not all) without losing original information. The number of concepts in different concept reducts of a formal context may vary, where the concept reducts with the fewest number of formal concepts indicate that all binary relations can be recovered by the fewest formal concepts. This greatly simplifies the knowledge representation. We first define the bases and the cardinality of a formal context, the most expanded concept reducts and the most compressed concept reducts, and investigate their properties. Then, we explore the relationship between the bases of formal context and the set of object concepts (the set of attribute concepts), and provide an equivalent condition for when the set of object concepts (the set of attribute concepts) serves as a basis. Finally, we present the range of the cardinality of a formal context from the perspectives of the set dimension and the Ferrers dimension of a formal context.

In incomplete contexts, SE-ISI concepts contain abundant uncertain information, but not all SE-ISI concepts are necessary. This paper studies the theories and methods of SE-ISI concept reduction under different situations in incomplete contexts. First, SE-ISI concept reductions preserving positive information, generalized positive information and relation are defined, respectively. Then, the relationships between the three types of SE-ISI concept reduction are analyzed. Moreover, by introducing the SE-ISI representative concept matrix, the methods of obtaining three types of SE-ISI concept reduction are given. Finally, the characteristics and relationships of SE-ISI concepts under three types of SE-ISI concept reduction are discussed from the perspective of SE-ISI representative concept matrix.

A formal context is the data base and starting point for research in formal concept analysis. In order to describe the binary relations satisfying reflexivity and symmetry in a formal context, this paper firstly defines the symmetric formal context, and studies the characteristics of the concept lattice, the object concept, the attribute concept, the join irreducible concept and the meet irreducible concept of the symmetric formal context. Then, we propose the symmetric concept and prove that the symmetric concept set is a concept reduct of symmetric formal context. Finally, from the perspective of conflict analysis and social network, we show the practical semantics and application of the symmetric formal context, the symmetric concept and the symmetric concept set, respectively.

This paper extends the concept reduction in formal context to formal decision context, and studies the concept reduction preserving antecedent information of rules in weakly consistent formal decision context. Firstly, the conditional subcontext is compressed to form the antecedent context by only considering the binary relation in the antecedent of the rules of formal decision context. Then, we propose the concept reduction preserving antecedent information of rules for the antecedent context, and give the judgment theorem of concept consistent set by representative concept matrix. Finally, we classify all formal concepts into three parts according to the role of each formal concept in concept reduction, and discuss the concept characteristics of three types of formal concepts from the perspective of the minimal representative concept matrix.

Concept reduction preserving binary relation is a new reduction theory in formal concept analysis, it can reduce the number of concepts without losing the original information. This paper studies the concept reduction of property-oriented concept lattices, this newly proposed reduction can preserve the complementary binary relation. First, the definition of property-oriented concept reduction is given. Then, the approach to obtain concept reduction and characteristics of property-oriented concept are given from the perspective of POC representative concept matrix. Moreover, the relationship between property-oriented concept reduction and object-oriented concept reduction are discussed. Finally, the application of property-oriented concept reduction in knowledge space theory is given.

Attribute reduction, as a very important branch of formal concept analysis, is equally important in the three-way concept analysis. Based on object-induced concept lattices, attribute reduction theory which preserve OE-consistency of formal decision contexts is proposed, which enriches the reduction theory of three-way concept analysis. Firstly, OE-consistent set and OE-reduct of formal decision contexts are defined, and the attributes are classified into three categories according to their characteristics. Then, it is pointed out the essence of OE-reduct is minimal OE-consistent set, and several judgement theorems of OE-consistent set are given. By studying the necessary and sufficient conditions of OE-consistent set, judgement theorem of OE-reduct is obtained. Finally, the definition of OE-discernibility matrix and OE-discernibility function are given, and the method of calculating OE-reduct by using OE-discernibility matrix and OE-discernibility function is given.