Concepts with negative-values and corresponding concept lattices

Formal Concept Analysis (FCA) provides a special method for extracting concepts from binary relations. A concept consists of the extent (objects the concept covers) and the intent (attributes describing the concept). From the propositional logical point of view, the intent of each concept only consists of attributes, and hence the classical FCA-concepts cannot describe negations of attributes. Therefore, it is necessary to extend the classical FCA-concepts to improve the expressiveness of concepts. In this paper, we take the negation into consideration in the process of constructing concepts, and hence obtain new extended concepts. Compared with the classical FCA-concepts, the extended concepts are more expressive. Let ℒ(R) and ℒ¬(R) represent respectively the concept lattice and the concept lattice with negative-values of a binary relation, and we prove that there is an infimum-preserving order-embedding map from ℒ(R) to ℒ¬(R), and conversely, there is a supremum-preserving order-preserving map from ℒ¬(R) to ℒ(R).