The Maximal Relation Based on A Given Relation Schema and Its Concept Lattice

Formal concept analysis (FCA) is a valid tool for data mining and knowledge discovery, which identifies concept lattices from binary relations. Given a nonempty finite set A of binary attributes, one obtains a maximal binary relation R based on a relation schema S(A). Firstly, we analyze concepts in R and the concept lattice L(R), and there are some important results as follows: for any two concepts in R, the union of their intents is an intent of some concept in R, and further the intent of their supremum is the union of their intents; for any two concepts in R, if one of them is not a sub-concept or super-concept of the other one, then the union of their extents is not an extent of any concept in R; L(R) is a complemented distributive lattice. Secondly, we provide the structural connection between L(R) and L(R): for any relation R based on S(A), there is a supremum-preserving order-embedding map from L(R) to L(R), and conversely, there is an infimumpreserving order-preserving map from L(R) to L(R), which is generally not a surjective homomorphism. Thirdly, we propose two algorithms to extract concepts in R from L(R), which are respectively based on intents and extents of concepts, and prove their soundness. These results have already been used to analyze the data in architectual engineering and medical science.