Constructing three-way concept lattices based on apposition and subposition of formal contexts

Three-way concept analysis provides a new model to make three-way decisions. Its basic structure can be shown by the three-way concept lattices. Thus, how to construct three-way concept lattices is an important issue in the three-way concept analysis. This paper proposes approaches to create the three-way concept lattices of a given formal context. First, we can transform the given formal context and its complementary context into new formal contexts which are isomorphic to the given formal context and its complementary context respectively. And then, Type I-combinatorial context and Type II-combinatorial context are defined, which are apposition and subposition of these new formal contexts, respectively. Second, we prove that the concept lattice of Type I-combinatorial context is isomorphic to object-induced three-way concept lattice and the concept lattice of Type II-combinatorial context is isomorphic to attribute-induced three-way concept lattice of the given formal context. And then, the approaches of creating the three-way concept lattices are proposed based on the concept lattices of Type I-combinatorial context and Type I-combinatorial context. Finally, we give the corresponding algorithms of constructing three-way concept lattices based on the above approaches and conduct several experiments to illustrate the efficient of proposed algorithms.