On The Object Enlargement Of Concept Lattice

This paper studies the enlargement of concept lattice in classical and fuzzy contexts.In the first part, we introduce the background, content and the primary methods of concept lattice research. In the second part, we discuss the object enlargement in the form of classical context, which means that finding the maximal object set such that (G,A,R_G)and(F,A,R_F) is isomorphic. And we will deal with the consistency of this enlargement. We discuss in the third part the exact enlargement and the approximate enlargement in a concept lattice by means of the similarity relation in the form of fuzzy context. Beholavek studied the similarity relations between two fuzzy concept lattices L(O,A, R₁) and L(O,A,R₂). Inspired by his discussion, this chapter studies the similarity relation between the fuzzy context (G, A, R_G) and its extracontext (F,A,R_F).Two new concepts of the extendable subcontext and unextendable subcontext are introduced, and the similarity relations between the fuzzy context(G,A,R_G)and(F,A,R_F)are defined respectively in the two case. Then we propose the concepts of the exact enlargement and the approximate enlargement in fuzzy concept lattices. The exact enlargement is the maximal objects set such that L(G, A,R_G) and L(F, A,R_F) are isomorphic. In other words, the similarity between L (G, A, R_G) and L (F, A, R_F)equal to 1. The approximate enlargement is the maximal objects set such that the similarity between L (G, A, R_G) and L (F, A, R_F) is not less than a given value. In the forth part, we studies the enlargement of a fuzzy concept lattice in a fuzzy context by employing the variable precision concept under various precisions. The cut-context is studied. It is proved that the variable precision concept is isomorphic to the concept lattice induced by the cut-context in the same precision. Thus, we can convert the enlargement of a variable precision concept into the enlargement of the concept lattice induced by the cut-context in the same precision. Then we can use the method of article [45] to obtain all of the enlargements in the fuzzy concept lattices. In the last part, we summarize the main results of this paper and propose the foreground of this study.