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

Abstract

Formal concept analysis (FCA) is a valid tool for data mining and knowledge discovery,<br />which identifies concept lattices from binary relations. Given a nonempty finite set $A$ of binary attributes, one obtains a maximal binary relation R^max based on a relation schema S(A). Firstly, we analyze concepts in R^max and the concept lattice L(R^max, and there are some important results as follows: for any two concepts in R^max, the union of their intents is an intent of some concept in R^max, and further the intent of their supremum is the union of their intents;<br />for any two concepts in R^max, 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^max; L(R^max) is a complemented distributive lattice. Secondly, we provide the structural connection between L(R) and&nbsp;L(R^max: for any relation R based on S(A), there is a supremum-preserving order-embedding map from&nbsp;L(R) to&nbsp;L(R^max), and conversely, there is an infimum-preserving order-preserving map from L(R^max) to L(R), which is generally not a surjective homomorphism.<br />Thirdly, we propose two algorithms to extract concepts in R from L(R^max), 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