Quotient lattice and incremental construction of concept lattices

Concept lattice which is the core data structure in formal concept analysis has proved to be a useful tool in many applied domains: machine learning, software engineering and information retrieval, etc. The problem of constructing concept lattices is extensively studied and among various kinds of algorithms published so far, incremental ones which incrementally update existing concept lattice with the inserted new object are of comparative advantage in execution efficiency. This article, on the basis of further study on incremental algorithms, develops quotient lattice theory and uses it to describe the relationship between original concept lattice and the new one. Furthermore, through construction of quotient lattice, a conclusion that all modified nodes and new nodes form a complete lattice with the same structure as the one formed by those corresponding ones in new concept lattice is drawn, which is helpful to deeply understanding of the inherent characteristics of generators and provides theoretical foundation for further improvement of such kind of algorithms.