Bases and cardinality of formal contexts and their associated properties

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.