Adjunctive three-way concepts from positive and negative concepts in lattice-valued formal contexts

Three-way concept analysis is an integration of formal concept analysis and three-way decision, which has been found wide applications in knowledge representation, cognitive computing and decision sciences. There have been already a huge number of literature on theory and applications on such topics. Most researches on formal concept analysis focus, independently, on specific scenarios such as Boolean formal contexts and fuzzy formal contexts. In this paper, a unified framework for three-way concept analysis is built in lattice-valued formal contexts. The notions of positive concept and negative concept are given in a lattice-valued formal context based on an overlapping function, one of significant operators for information fusion. Two pairs of object induced adjunctive three-way concept operators and attribute induced adjunctive three-way concept operators are introduced. Their properties have been fully investigated. With those operators, the algebraic structures of adjunctive three-way concepts, their extensions and intensions have been explored. Specifically, the local structure within pairs of sub-concept and sup-concept has been finely characterized. The derived results provide ones a way to construct the complete lattice of a lattice-valued formal context or its sublattice by determining all pairs of sub-concept and sup-concept provided that an object fuzzy set or an attribute fuzzy set is given. A case analysis on a real world dataset from UCI repository is presented to verify the rationality and significance of obtained results.