Subdirect Decomposition of Contexts into Subdirectly Irreducible Factors

The size of a concept lattice may increase exponentially with the size of the context. When the number of nodes is too large, it becomes very difficult to generate and study such a concept lattice. A way to avoid this problem is to break down the lattice into small parts. In the subdirect decomposition, the small parts are factor lattices which are meaningful in the Formal Concept Analysis (FCA) setting. In this paper a walkthrough from a finite reduced context to its subdirect decomposition into subdirectly irreducible subcontexts and factors is given. The decomposition can be reached using three different points of view, namely factor lattices, arrow relations and compatible subcontexts. The approach is mainly algebraic since it is based on abstract lattice theory, except for the last point which is inherited from FCA. We also propose a polynomial algorithm to generate the decomposition of an initial context into subcontexts. Such a procedure can be extended to conduct an interactive exploration and mining of large contexts, including the generation of few concepts and their neighborhood.