On Rough Concept Lattices

Formal concept analysis and rough set theory provide two different methods for data analysis and knowledge processing. Given a context K, one can get the concept lattice L(K) in Wille's sense and the object-oriented rough concept lattice RO-L(K) (resp., attribute-oriented RA-L(K)). We study relations of the three kinds of lattices and their properties from the domain theory point of view. The concept of definable sets is introduced. It is proved that the family Def (K) of the definable sets in set-inclusion order is a complete sublattice of RO-L(K) and is a complete field of sets under some reasonable conditions. A necessary and sufficient condition for Def (K) to be equal to RO-L(K) is given. A necessary and sufficient condition is also given for the complete distributivity of RO-L(K). We also study algebraicity of RO-L(K) and several sufficient conditions are given for RO-L(K) to be algebraic.