L-concept analysis with positive and negative attributes

We describe an extension of formal fuzzy concept analysis allowing a user to choose which attributes are viewed as positive and which are viewed as negative. The two sets are then handled using a combination of previously studied antitone concept-forming operators and isotone concept-forming operators, respectively. The two main outputs of formal concept analysis, namely concept lattices and attribute implications, in the setting of positive and negative attributes are presented. An analogy of the main theorem of concept lattices and a relationship between the new concept lattice and the previously studied concept lattices is showed. We introduce basic syntactic and semantic notions for attribute implications called fuzzy containment implications. We consider two settings, one where the sets of positive and negative attributes are crisp sets, and a generalization, where the two sets are fuzzy sets.