Three-way granular computing, rough sets, and formal concept analysis.

Three-way granular computing is a paradigm of thinking and information processing in three granules. We introduce a framework of three-way granular computing based on a hexagon induced by a trisection of a set. This enables us to unify rough-set concept analysis and formal concept analysis. For an object granule (i.e., a subset of a universal set of objects), we construct three attribute granules. Conversely, for an attribute granule (i.e., a subset of a universal set of attributes), we construct three object granules. Rough-set concept analysis uses one kind of trisections to study four types of disjunctive formal concepts and the associated concept lattices. Formal concept analysis uses another kind of trisections to study four types of conjunctive formal concepts and the associated concept lattices. The eight concept lattices can be grouped into two classes such that four lattices in each class are isomorphic.