Three-way concept lattice from adjunctive positive and negative concepts

Three-way concept lattices (TCLs) have been widely explored due to their clear hierarchical structures, concise visual description and good interpretability. In contrast to classic formal contexts, lattice-valued fuzzy contexts exhibit great capability in describing and representing concepts with uncertainty. Different from conventional approaches to research of TCLs, this paper focuses on investigating the algebraic structure and properties of three-way concept lattice (TCL) stemmed from the positive concept lattice and negative concept lattice in a lattice-valued formal context. Several associated concept lattices such as the Cartesian product of positive concept lattice and negative lattice (i.e., pos-neg lattice), lattices induced from the partition of the pos-neg lattice, and their relationship are explored. Specifically, the isomorphism, embedding and order-preserving mappings between them are built. The quotient set of pos-neg lattice when being defined a specific equivalence relation on it is a complete lattice and each equivalence class is a lower semi-lattice. It is further declared that the structure of TCL is intrinsically and determined wholly by the pos-neg lattice. A practical application of the built theory of TCL is provided to sort alternatives in multi-criteria decision making.