The structure theorem of three-way concept lattice

Three-way decision (3WD) is a widely used and studied mathematical theory that generalizes the thinking norm of tri-level in cognitive learning, problem solving, and information processing. By utilizing the negative information contained in data, three-way concept lattice (3WCL) developed 3WD in Formal Concept Analysis and has been applied in various applications such as conflict analysis, role based access control, knowledge discovery, concept learning, and medical diagnose. However, the connections between 3WCL and classical concept lattices have not received its deserved attention. To this end, first, this paper proved that 3WCL is exactly the minimal closure system containing both concept lattice and complementary concept lattice, and classified three-way concepts into four categories. Second, this paper proved the structure theorem of 3WCL that characterizes mathematically the relationships between concept lattice, complementary concept lattice and 3WCL as two isomorphisms. Third, this paper presented several applications of the structure theorem to reveal its essentiality in discussing the properties of 3WCL. Finally, some problems that are not involved in the structure theorem are also discussed.