Links between Modular Decomposition of Concept Lattice and Bimodular Decomposition of a Context

This paper is a preliminary attempt to study how modular and bimodular decomposition, used in graph theory, can be used on contexts and concept lattices in formal concept analysis (FCA). In a graph, a module is a set of vertices defined in term of behaviour with respect to the outside of the module: All vertices in the module act with no distinction and can be replaced by a unique vertex, which is a representation of the module. This definition may be applied to concepts of lattices, with slighty modifications (using order relation instead of adjacency). One can note that modular decomposition is not well suited for bipartite graphs. For example, every bipartite graph corresponding to a clarified context is trivially prime (not decomposable w.r.t modules). In [4], authors have introduced a decomposition dedicaced to bipartite graph, called the bimodular decomposition. In this paper, we show how modular decomposition of lattices and bimodular decomposition of contexts interact. These results may be used to improve readability of a Hasse diagram.